Answers to some of 4clojure questions. Please note that the original 4clojure.com website has been discontinued, apparently 4clojure.oxal.org is the new site to check-out.
It looks like there are some broken questions or gaps between questions sometimes (for the original 4clojure.com site).
I did couple of problems and I believe that most of my current basic issues are related to:
- writing non-idiomatic code (tendency to think in Java)
- Graph manipulations/traversal in a pure functional way.
- Do “real” Clojure books reading, as reading is fundamental.
- I have a basic Compojure application that I should finish:
I’ve been looking at it about once a month for a while… (Compojure, Korma, Java JCE, etc.)
- Need to master about macros well enough.
- Need to explore stuff such as core.async, transducers, transients, etc.
- Need to do more puzzles, as my background isn’t pure computer science or engineering.
Nothing but the Truth
;; This is a clojure form.
;; Enter a value which will make the form evaluate to true.
;; Don't over think it! If you are confused, see the
;; <a href='/directions'>getting started</a> page.
;; Hint: true is equal to true.
;; (= __ true)
(= true true)Simple Math
;; <p>If you are not familiar with <a href='http://en.wikipedia.org/wiki/Polish_notation'>
;; polish notation</a>
;; , simple arithmetic might seem confusing.</p><p><strong>Note:</strong>
;; Enter only enough to fill in the blank
;; (in this case, a single number) - do not retype the whole problem.</p>
;; (= (- 10 (* 2 3)) __)
(= (- 10 (* 2 3)) 4)Intro to Strings
;; Clojure strings are Java strings.
;; This means that you can use any of the Java string methods on Clojure strings.
;; (= __ (.toUpperCase "hello world"))
(= "HELLO WORLD" (.toUpperCase "hello world"))Intro to Lists
;; Lists can be constructed with either a function or a quoted form.
;; (= (list __) '(:a :b :c))
(= (list :a :b :c) '(:a :b :c))Lists: conj
;; When operating on a list, the conj function will return
;; a new list with one or more items "added" to the front.
;; (= __ (conj '(2 3 4) 1))
(= '(1 2 3 4) (conj '(2 3 4) 1))Intro to Vectors
;; Vectors can be constructed several ways. You can compare them with lists.
;; (= [__] (list :a :b :c) (vec '(:a :b :c)) (vector :a :b :c))
(= [:a :b :c] (list :a :b :c) (vec '(:a :b :c)) (vector :a :b :c))Vectors: conj
;; When operating on a Vector, the conj function will return a new vector with one or
;; more items "added" to the end.
;; (= __ (conj [1 2 3] 4))
(= [ 1 2 3 4] (conj [1 2 3] 4))Intro to Sets
;; Sets are collections of unique values.
;; (= __ (set '(:a :a :b :c :c :c :c :d :d)))
(= #{:a :b :c :d} (set '(:a :a :b :c :c :c :c :d :d)))Sets: conj
;; When operating on a set, the conj function returns a new set with one or more keys
;; "added".
;; (= #{1 2 3 4} (conj #{1 4 3} __))
(= #{1 2 3 4} (conj #{1 4 3} 2))Intro to Maps
;; Maps store key-value pairs. Both maps and keywords can be used as lookup functions.
;; Commas can be used to make maps more readable, but they are not required.
;; (= __ ((hash-map :a 10, :b 20, :c 30) :b))
(= 20 ((hash-map :a 10, :b 20, :c 30) :b))Maps: conj
;; When operating on a map, the conj function returns a new map with one or more
;; key-value pairs "added".
;; (= {:a 1, :b 2, :c 3} (conj {:a 1} __ [:c 3]))
(= {:a 1, :b 2, :c 3} (conj {:a 1} {:b 2} [:c 3]))Intro to Sequences
;; All Clojure collections support sequencing. You can operate on sequences with
;; functions like first, second, and last.
;; (= __ (first '(3 2 1)))
(= 3 (first '(3 2 1)))Sequences: rest
;; The rest function will return all the items of a sequence except the first.
;; (= __ (rest [10 20 30 40]))
(= [20 30 40] (rest [10 20 30 40]))Intro to Functions
;; Clojure has many different ways to create functions.
;; (= __ ((fn add-five [x] (+ x 5)) 3))
(= 8 ((fn add-five [x] (+ x 5)) 3))Double Down
;; Write a function which doubles a number.
;; (= (__ 2) 4)
(defn double-num [n]
(* n 2))
(clojure.test/testing
"Write a function which doubles a number."
(clojure.test/is (and
(= (double-num 2) 4)
(= (double-num 3) 6)
(= (double-num 11) 22)
(= (double-num 7) 14))))Hello World
;; Write a function which returns a personalized greeting.
;; (= (__ "Dave") "Hello, Dave!")
(defn greet [someone]
(format "Hello, %s!" someone))
(clojure.test/testing
"Write a function which returns a personalized greeting."
(clojure.test/is (and
(= (greet "Dave") "Hello, Dave!")
(= (greet "Jenn") "Hello, Jenn!")
(= (greet "Rhea") "Hello, Rhea!"))))Sequences: map
;; The map function takes two arguments: a function (f) and a sequence (s).
;; Map returns a new sequence consisting of the result of applying f to each item of s.
;; Do not confuse the map function with the map data structure.
;; (= __ (map #(+ % 5) '(1 2 3)))
(= '( 6 7 8) (map #(+ % 5) '(1 2 3)))Sequences: filter
;; The filter function takes two arguments: a predicate function (f) and a sequence (s).
;; Filter returns a new sequence consisting of all the items of s for which (f item)
;; returns true.
;; (= __ (filter #(> % 5) '(3 4 5 6 7)))
(= '(6 7) (filter #(> % 5) '(3 4 5 6 7)))Last Element
;; Write a function which returns the last element in a sequence.
;; Restrictions (please don't use these function(s)): last
;; (= (__ [1 2 3 4 5]) 5)
(defn last-elem [[n & more]]
(if more
(recur more)
n))
(clojure.test/testing
"Write a function which returns the second to last
element from a sequence."
(clojure.test/is (and
(= (last-elem [1 2 3 4 5]) 5)
(= (last-elem '(5 4 3)) 3)
(= (last-elem ["b" "c" "d"]) "d"))))Penultimate Element
;; Write a function which returns the second to last element from a sequence.
(defn second-to-last [[x & xs]]
(if (= 1 (count xs)) x
(recur xs)))
(clojure.test/testing
"Write a function which returns the second to last element from a sequence."
(clojure.test/is (and
(= (second-to-last (list 1 2 3 4 5)) 4)
(= (second-to-last ["a" "b" "c"]) "b")
(= (second-to-last [[1 2] [3 4]]) [1 2]))))Nth Element
;; Write a function which returns the Nth element from a sequence.
;; Restrictions (please don't use these function(s)): nth
;; (= (__ '(4 5 6 7) 2) 6)
(defn nth-element [[x & xs] idx]
(if (= idx 0) x
(recur xs (dec idx))))
(= (nth-element '(4 5 6 7) 2) 6)Count a Sequence
;; Write a function which returns the total number of elements in a sequence.
;; Restrictions (please don't use these function(s)): count
(defn count-seq [xs]
(reduce (fn [sum _] (inc sum)) 0 xs))
(and
(= (count-seq '(1 2 3 3 1)) 5)
(= (count-seq "Hello World") 11)
(= (count-seq [[1 2] [3 4] [5 6]]) 3)
(= (count-seq '(13)) 1)
(= (count-seq '(:a :b :c)) 3))Reverse a Sequence
;; Write a function which reverses a sequence.
;; Restrictions (please don't use these function(s)): reverse, rseq
;; (= (__ [1 2 3 4 5]) [5 4 3 2 1])
(defn reverse-seq [xs]
(into '() xs))
(clojure.test/testing
"Write a function which reverses a sequence."
(clojure.test/is (and
(= (reverse-seq [1 2 3 4 5]) [5 4 3 2 1])
(= (reverse-seq (sorted-set 5 7 2 7)) '(7 5 2))
(= (reverse-seq [[1 2][3 4][5 6]]) [[5 6][3 4][1 2]]))))Sum It All Up
;; Write a function which returns the sum of a sequence of numbers.
;; (= (__ [1 2 3]) 6)
(defn sum-xs [xs]
(reduce + xs))
(clojure.test/testing
"Write a function which returns the sum of a sequence of numbers."
(clojure.test/is (and
(= (sum-xs [1 2 3]) 6)
(= (sum-xs (list 0 -2 5 5)) 8)
(= (sum-xs #{4 2 1}) 7)
(= (sum-xs '(0 0 -1)) -1)
(= (sum-xs '(1 10 3)) 14))))Find the odd numbers
;; Write a function which returns only the odd numbers from a sequence.
;; (= (__ #{1 2 3 4 5}) '(1 3 5))
(defn odd-numbers [xs]
(filter odd? xs))
(clojure.test/testing
"Only odd numbers."
(clojure.test/is (= (odd-numbers #{1 2 3 4 5}) '(1 3 5))))Fibonacci Sequence
;; Write a function which returns the first X fibonacci numbers.
;; (= (__ 3) '(1 1 2))
(defn fib [n]
{:pre [(pos? n)]}
(letfn [(fibonacci [a b]
(lazy-seq
(cons (+ a b) (fibonacci b (+ a b)))))]
(take n (cons 1 (fibonacci 0 1)))))
(clojure.test/testing
"Write a function which returns the first X fibonacci numbers."
(clojure.test/is (and
(= (fib 3) '(1 1 2))
(= (fib 6) '(1 1 2 3 5 8))
(= (fib 8) '(1 1 2 3 5 8 13 21)))))
Palindrome Detector
;; Write a function which returns true if the given sequence is a palindrome.
;; Hint: "racecar" does not equal '(\r \a \c \e \c \a \r)
(defn palindrome? [xs]
(every? #(true? %) (map #(= %1 %2) xs (reverse xs))))
(and
(false? (palindrome? '(1 2 3 4 5)))
(true? (palindrome? "racecar"))
(true? (palindrome? [:foo :bar :foo]))
(true? (palindrome? '(1 1 3 3 1 1)))
(false? (palindrome? '(:a :b :c))))
Flatten a Sequence
;; Write a function which flattens a sequence.
;; Restrictions (please don't use these function(s)): flatten
(defn my-flatten [xs]
(lazy-seq
(reduce (fn --internal-flatten [col v]
(if (sequential? v)
(reduce --internal-flatten col v)
(conj col v)))
[]
xs)))
(and (= (my-flatten '((1 2) 3 [4 [5 6]])) '(1 2 3 4 5 6))
(= (my-flatten ["a" ["b"] "c"]) '("a" "b" "c"))
(= (my-flatten '((((:a))))) '(:a)))Get the Caps
;; Write a function which takes a string and returns a new string containing only
;; the capital letters.
;; (= (__ "HeLlO, WoRlD!") "HLOWRD")
(defn only-caps [s]
(reduce str (filter #(Character/isUpperCase %1) s)))
(clojure.test/testing
"Write a function which takes a string and returns
a new string containing only the capital letters."
(clojure.test/is (and
(= (only-caps "HeLlO, WoRlD!") "HLOWRD")
(empty? (only-caps "nothing"))
(= (only-caps "$#A(*&987Zf") "AZ"))))
Compress a Sequence
;; Write a function which removes consecutive duplicates from a sequence.
;; maybe more elegant and idiomatic, do not thing it is faster than
;; the first reduce version though but did not time it.
(defn del-consecutive-dups [col]
(mapcat set (#(partition-by identity %1) col)))
(and (= (apply str (del-consecutive-dups "Leeeeeerrroyyy")) "Leroy")
(= (del-consecutive-dups [1 1 2 3 3 2 2 3]) '(1 2 3 2 3))
(= (del-consecutive-dups [[1 2] [1 2] [3 4] [1 2]]) '([1 2] [3 4] [1 2])))Pack a Sequence
;; Write a function which packs consecutive duplicates into sub-lists.
;; Took more than few mins for something so simple
;; I'm not fluent yet with group-by vs split-width vs partition
(defn partition-dups [col]
(partition-by identity col))
(and
(= (partition-dups [1 1 2 1 1 1 3 3]) '((1 1) (2) (1 1 1) (3 3)))
(= (partition-dups [:a :a :b :b :c]) '((:a :a) (:b :b) (:c)))
(= (partition-dups [[1 2] [1 2] [3 4]]) '(([1 2] [1 2]) ([3 4]))))
Duplicate a Sequence
;; Write a function which duplicates each element of a sequence.
(defn dup-each-item [xs]
(reduce #(apply conj %1 (list %2 %2)) [] xs))
(and
(= (dup-each-item [1 2 3]) '(1 1 2 2 3 3))
(= (dup-each-item [:a :a :b :b]) '(:a :a :a :a :b :b :b :b))
(= (dup-each-item [[1 2] [3 4]]) '([1 2] [1 2] [3 4] [3 4]))
(= (dup-each-item [[1 2] [3 4]]) '([1 2] [1 2] [3 4] [3 4])))Replicate a Sequence
;; Write a function which replicates each element
;; of a sequence a variable number of times.
(defn replicate-each-item [col n-times]
(mapcat #(repeat n-times %1) col))
(and (= (replicate-each-item [1 2 3] 2) '(1 1 2 2 3 3))
(= (replicate-each-item [:a :b] 4) '(:a :a :a :a :b :b :b :b))
(= (replicate-each-item [4 5 6] 1) '(4 5 6))
(= (replicate-each-item [[1 2] [3 4]] 2) '([1 2] [1 2] [3 4] [3 4]))
(= (replicate-each-item [44 33] 2) [44 44 33 33]))Implement range
;; Write a function which creates a list of all integers in a given range.
;; Restrictions (please don't use these function(s)): range
;; (= (__ 1 4) '(1 2 3))
(defn find-range [start end]
(take (- end start) (iterate inc start)))
(clojure.test/testing
"Write a function which creates a list of all integers in a given range."
(clojure.test/is (and (= (find-range 1 4) '(1 2 3))
(= (find-range -2 2) '(-2 -1 0 1))
(= (find-range 5 8) '(5 6 7)))))Local bindings
;; Clojure lets you give local names to values using the special let-form.
;; (= __ (let [x 5] (+ 2 x)))
;; (= __ (let [x 3, y 10] (- y x)))
;; (= __ (let [x 21] (let [y 3] (/ x y))))
(clojure.test/testing
"Clojure lets you give local names to values using the special let-form."
(clojure.test/is (and (= 7 (let [x 5] (+ 2 x)))
(= 7 (let [x 3 y 10] (- y x)))
(= 7 (let [x 21] (let [y 3] (/ x y)))))))
Let it Be
;; Can you bind x, y, and z so that these are all true?
;; (= 10 (let __ (+ x y)))
;; (= 4 (let __ (+ y z)))
;; (= 1 (let __ z))
(clojure.test/testing
"Can you bind x, y, and z so that these are all true?"
(clojure.test/is (and
(= 10 (let [x 7 y 3 z 1] (+ x y)))
(= 4 (let [x 7 y 3 z 1] (+ y z)))
(= 1 (let [x 7 y 3 z 1] z)))))Regular Expressions
;; Regex patterns are supported with a special reader macro.
(= "ABC" (apply str (re-seq #"[A-Z]+" "bA1B3Ce ")))Maximum value
;; Write a function which takes a variable number of parameters
;; and returns the maximum value.
;; Restrictions (please don't use these function(s)): max, max-key
(defn max-value [x & xs]
(reduce (fn [x y] (if (pos? (.compareTo y x)) y x)) x xs))
(clojure.test/testing
"Write a function which takes a variable number of
parameters and returns the maximum value."
(clojure.test/is (and
(= (max-value 1 8 3 4) 8)
(= (max-value 30 20) 30)
(= (max-value 45 67 11) 67))))
Interleave Two Seqs
;; Write a function which takes two sequences and
;; returns the first item from each, then the second item
;; from each, then the third, etc.
;; Restrictions (please don't use these function(s)): interleave
(defn my-interleave [x1 x2]
(lazy-seq
(when-not (or (empty? x1) (empty? x2))
(cons (first x1) (cons (first x2)
(my-interleave (rest x1) (rest x2)))))))
(and
(= (my-interleave [1 2 3] [:a :b :c]) '(1 :a 2 :b 3 :c))
(= (my-interleave [1 2] [3 4 5 6]) '(1 3 2 4))
(= (my-interleave [1 2 3 4] [5]) [1 5])
(= (my-interleave [30 20] [25 15]) [30 25 20 15]))
Interpose a Seq
;; Write a function which separates the items
;; of a sequence by an arbitrary value.
;;
;; Restrictions (please don't use these function(s)):
;; interpose
(defn my-interpose [delimiter [x & more]]
(lazy-seq
(when x
(if more
(cons x (cons delimiter (my-interpose delimiter more)))
(cons x nil)))))
(and
(= (my-interpose 0 [1 2 3]) [1 0 2 0 3])
(= (apply str (my-interpose ", " ["one" "two" "three"])) "one, two, three")
(= (my-interpose :z [:a :b :c :d]) [:a :z :b :z :c :z :d]))Drop Every Nth Item
;; Write a function which drops every Nth item from a sequence.
;; simplistic approach no accumulator in a loop or similar
;; try to write more idiomatic code first.
(defn my-drop-every [col n]
(when col
(lazy-cat (take (dec n) col) (my-drop-every (nthnext col n) n))))
(and
(= (my-drop-every [1 2 3 4 5 6 7 8] 3) [1 2 4 5 7 8])
(= (my-drop-every [:a :b :c :d :e :f] 2) [:a :c :e])
(= (my-drop-every [1 2 3 4 5 6] 4) [1 2 3 5 6]))Factorial Fun
;; Write a function which calculates factorials.
(defn factorial [n]
(reduce * (range 1 (inc n))))
(clojure.test/testing
"Write a function which calculates factorials."
(clojure.test/is (and
(= (factorial 1) 1)
(= (factorial 3) 6)
(= (factorial 5) 120)
(= (factorial 8) 40320))))
Reverse Interleave
;; Write a function which reverses the interleave
;; process into x number of subsequences.
(defn reverse-interleave [xs n]
(letfn [(stepper [col nb-items step limit]
(when (pos? limit)
(cons (take nb-items (take-nth step col))
(stepper (next col) nb-items step (dec limit)))))]
(stepper xs (/ (count xs) n) n n)))
(and (= (reverse-interleave [1 2 3 4 5 6] 2) '((1 3 5) (2 4 6)))
(= (reverse-interleave (range 9) 3) '((0 3 6) (1 4 7) (2 5 8)))
(= (reverse-interleave (range 10) 5) '((0 5) (1 6) (2 7) (3 8) (4 9))))Rotate Sequence
;; Write a function which can rotate a sequence in either direction.
(defn rotate-xs [dir xs]
(let [ln (count xs)]
(if (pos? dir)
(take ln (drop dir (cycle xs)))
(take ln (drop (- ln (mod (* dir -1) ln)) (cycle xs))))))
(and (= (rotate-xs 2 [1 2 3 4 5]) '(3 4 5 1 2))
(= (rotate-xs -2 [1 2 3 4 5]) '(4 5 1 2 3))
(= (rotate-xs 6 [1 2 3 4 5]) '(2 3 4 5 1))
(= (rotate-xs 1 '(:a :b :c)) '(:b :c :a))
(= (rotate-xs -4 '(:a :b :c)) '(:c :a :b)))Intro to Iterate
;; The iterate function can be used to produce an infinite lazy sequence.
;; (= __ (take 5 (iterate #(+ 3 %) 1)))
(= '(1 4 7 10 13) (take 5 (iterate #(+ 3 %) 1)))Flipping out
;; Write a higher-order function which flips the order
;; of the arguments of an input function.
(defn flip-args [f]
(fn [& args]
(apply f (reverse args))))
(clojure.test/testing
"Write a higher-order function which flips the order of the
arguments of an input function."
(clojure.test/is (and
(= 3 ((flip-args nth) 2 [1 2 3 4 5]))
(= true ((flip-args >) 7 8))
(= 4 ((flip-args quot) 2 8))
(= [1 2 3] ((flip-args take) [1 2 3 4 5] 3)))))Contain Yourself
;; The contains? function checks if a KEY is present in a
;; given collection.
;; This often leads beginner clojurians to use it
;; incorrectly with numerically indexed collections like vectors and lists.
(contains? #{4 5 6} 4)
(contains? [1 1 1 1 1] 1)
(contains? {4 :a 2 :b} 2)Intro to some
;; The some function takes a predicate function and a collection.
;; It returns the first logical true value of (predicate x)
;; where x is an item in the collection.
(= 6 (some #{2 7 6} [5 6 7 8]))
(= 6 (some #(when (even? %) %) [5 6 7 8]))Split a sequence
;; Write a function which will split a sequence into two parts.;;
;; Restrictions (please don't use these function(s)): split-at
;; Initial implementation used (vector (take n xs) (drop n xs)))
;; traverses twice the sequence...
(defn my-split-at [n xs]
((fn step [acc xs idx limit]
(if (= idx limit)
(conj [] acc (into [] xs))
(step (conj acc (first xs)) (next xs) (inc idx) limit)))
[] xs 0 n))
(clojure.test/testing
"Write a function which will split a sequence into two parts."
(clojure.test/is (and
(= (my-split-at 3 [1 2 3 4 5 6]) [[1 2 3] [4 5 6]])
(= (my-split-at 1 [:a :b :c :d]) [[:a] [:b :c :d]])
(= (my-split-at 2 [[1 2] [3 4] [5 6]]) [[[1 2] [3 4]] [[5 6]]]))))Split by Type
;; Write a function which takes a sequence consisting of items
;; with different types and splits them up into a set of
;; homogeneous sub-sequences. The internal order of each
;; sub-sequence should be maintained, but the sub-sequences
;; themselves can be returned in any order (this is why
;; 'set' is used in the test cases).
(defn type-partition [col]
(vals (group-by #(type %1) col)))
(and
(= (set (type-partition [1 :a 2 :b 3 :c])) #{[1 2 3] [:a :b :c]})
(= (set (type-partition [:a "foo" "bar" :b])) #{[:a :b] ["foo" "bar"]})
(= (set (type-partition [[1 2] :a [3 4] 5 6 :b])) #{[[1 2] [3 4]] [:a :b] [5 6]}))Advanced Destructuring
;; Problem 51
;;
;; Here is an example of some more sophisticated destructuring.
(= [1 2 [3 4 5] [1 2 3 4 5]] (let [[a b & c :as d] [1 2 3 4 5]] [a b c d]))Intro to Destructuring
;; Problem 52
;;
;; Let bindings and function parameter lists support destructuring.
(= [2 4]
(let [[a b c d e f g] (range)]
[c e]))Longest Increasing Sub-Seq
;; Given a vector of integers, find the longest consecutive sub-sequence
;; of increasing numbers. If two sub-sequences have the same length,
;; use the one that occurs first.
;; An increasing sub-sequence must have a length of 2 or greater to qualify.
;;
(defn lis [xs]
(letfn [(make-piles [col]
(reduce (fn [acc num]
(let [xs (last acc)]
(if (or (nil? xs) (<= num (peek xs)))
(conj acc [num])
(assoc acc (dec (count acc)) (conj xs num)))))
[] col))
(max-seq [piles]
(or (first (filter #(>= (count %1) 2) (sort-by count > piles))) '()))]
(-> (make-piles xs) (max-seq))))
(and (= (lis [1 0 1 2 3 0 4 5]) [0 1 2 3])
(= (lis [5 6 1 3 2 7]) [5 6])
(= (lis [2 3 3 4 5]) [3 4 5])
(= (lis [7 6 5 4]) []))Partition a Sequence
;; Write a function which returns a sequence of lists of x items each.
;; Lists of less than x items should not be returned.
;;
;; Restrictions (please don't use these function(s)): partition, partition-all
(defn my-partition [n c]
(lazy-seq
(when (>= (count c) n)
(cons (take n c) (my-partition n (nthnext c n))))))
(and
(= (my-partition 3 (range 9)) '((0 1 2) (3 4 5) (6 7 8)))
(= (my-partition 2 (range 8)) '((0 1) (2 3) (4 5) (6 7)))
(= (my-partition 3 (range 8)) '((0 1 2) (3 4 5))))Count Occurrences
(defn map-frequencies
"Map occurrences of numbers.
Should not use frequencies function."
[xs]
(reduce (fn [m i] (assoc m i (inc (m i 0)))) {} xs))
(clojure.test/testing
"Write a function which returns a map containing the number
of occurences of each distinct item in a sequence."
(clojure.test/is
(and
(= (map-frequencies [1 1 2 3 2 1 1]) {1 4, 2 2, 3 1})
(= (map-frequencies [:b :a :b :a :b]) {:a 2, :b 3})
(= (map-frequencies '([1 2] [1 3] [1 3])) {[1 2] 1, [1 3] 2}))))Find Distinct Items
;; Find Distinct Items
;; Difficulty: Medium
;; Topics: seqs core-functions
(defn only-distinct [col]
(reduce (fn [xs item] (if (some #(= item %1) xs) xs (conj xs item)))
[] col))
(clojure.test/testing
"Write a function which removes the duplicates from a sequence.
Order of the items must be maintained."
(clojure.test/is
(and
(= (only-distinct [1 2 1 3 1 2 4]) [1 2 3 4])
(= (only-distinct [:a :a :b :b :c :c]) [:a :b :c])
(= (only-distinct '([2 4] [1 2] [1 3] [1 3])) '([2 4] [1 2] [1 3]))
(= (only-distinct (range 50)) (range 50)))))Simple Recursion
;; Simple Recursion
;; Difficulty: Elementary
;;Topics: recursion
(clojure.test/testing
"A recursive function is a function which calls itself.
This is one of the fundamental techniques used in functional programming."
(clojure.test/is
(= '(5 4 3 2 1) ((fn foo [x] (when (> x 0) (conj (foo (dec x)) x))) 5))))Function Composition
;; Write a function which allows you to create function compositions.
;; The parameter list should take a variable number of functions,
;; and create a function applies them from right-to-left.
;;
;; Restrictions (please don't use these function(s)): comp
(defn compclj [& fs]
(fn [& args]
(reduce #(apply %2 (list %1)) args (reverse fs))))
(clojure.test/testing
"Write a function which allows you to create function compositions.
The parameter list should take a variable number of functions,
and create a function applies them from right-to-left."
(clojure.test/is
(and
(= [3 2 1] ((compclj rest reverse) [1 2 3 4]))
(= 5 ((compclj (partial + 3) second) [1 2 3 4])
(= true ((compclj zero? #(mod % 8) +) 3 5 7 9))
(= "HELLO" ((compclj #(.toUpperCase %) #(apply str %) take) 5 "hello world"))))))Juxtaposition
;; Take a set of functions and return a new function
;; that takes a variable number of arguments and
;; returns a sequence containing the result of
;; applying each function left-to-right to the argument list.
;;
;; Restrictions (please don't use these function(s)): juxt
(defn map-apply [& fs]
(fn [& args]
(map #(apply %1 args) fs)))
(and (= [21 6 1] ((map-apply + max min) 2 3 5 1 6 4))
(= ["HELLO" 5] ((map-apply #(.toUpperCase %) count) "hello"))
(= [2 6 4] ((map-apply :a :c :b) {:a 2, :b 4, :c 6, :d 8 :e 10})))Sequence Reductions
(defn my-reductions
([f col]
(my-reductions f (first col) (rest col)))
([f init col]
(cons
init
(lazy-seq
(if (empty? col)
nil
(my-reductions f
(apply f (list init (first col)))
(rest col)))))))
(clojure.test/testing
"Problem 60.
Write a function which behaves like reduce,
but returns each intermediate value of the reduction.
Your function must accept either two or three arguments,
and the return sequence must be lazy."
(clojure.test/is
(and (= (take 5 (my-reductions + (range))) [0 1 3 6 10])
(= (my-reductions conj [1] [2 3 4]) [[1] [1 2] [1 2 3] [1 2 3 4]])
(= (last (my-reductions * 2 [3 4 5])) (reduce * 2 [3 4 5]) 120))))Map Construction
(defn do-zipmap [ks vs]
(apply hash-map (interleave ks vs)))
(clojure.test/testing
"Problem 61.
Write a function which takes a vector of keys and
a vector of values and constructs a map from them.
Restrictions (please don't use these function(s)): zipmap."
(clojure.test/is
(and (= (do-zipmap [:a :b :c] [1 2 3]) {:a 1, :b 2, :c 3})
(= (do-zipmap [1 2 3 4] ["one" "two" "three"]) {1 "one", 2 "two", 3 "three"})
(= (do-zipmap [:foo :bar] ["foo" "bar" "baz"]) {:foo "foo", :bar "bar"}))))Re-implement Iterate
;; Given a side-effect free function f and an initial
;; value x write a function which returns an infinite
;; lazy sequence of x, (f x), (f (f x)), (f (f (f x))), etc.
(defn do-iterate [f x]
(cons x
(lazy-seq
(do-iterate f (f x)))))
(clojure.test/testing
"Given a side-effect free function f and an initial
value x write a function which returns an infinite
lazy sequence of x, (f x), (f (f x)), (f (f (f x))), etc."
(clojure.test/is
(and
(= (take 5 (do-iterate #(* 2 %) 1)) [1 2 4 8 16])
(= (take 100 (do-iterate inc 0)) (take 100 (range)))
(= (take 9 (do-iterate #(inc (mod % 3)) 1)) (take 9 (cycle [1 2 3]))))))Group a Sequence
;; Given a function f and a sequence s, write a function which returns a map.
;; The keys should be the values of f applied to each item in s.
;; The value at each key should be a vector of corresponding items
;; in the order they appear in s.
(defn do-group-by [f s]
(reduce (fn [m i]
(assoc m
(f i)
(conj (m (f i) []) i)))
{}
s))
(and
(= (do-group-by #(> % 5) [1 3 6 8]) {false [1 3], true [6 8]})
(= (do-group-by #(apply / %) [[1 2] [2 4] [4 6] [3 6]])
{1/2 [[1 2] [2 4] [3 6]], 2/3 [[4 6]]})
(= (do-group-by count [[1] [1 2] [3] [1 2 3] [2 3]])
{1 [[1] [3]], 2 [[1 2] [2 3]], 3 [[1 2 3]]}))Intro to Reduce
(clojure.test/testing
"Reduce takes a 2 argument function and an optional starting value.
It then applies the function to the first 2 items in the sequence
(or the starting value and the first element of the sequence).
In the next iteration the function will be called on the previous
return value and the next item from the sequence,
thus reducing the entire collection to one value.
Don't worry, it's not as complicated as it sounds."
(clojure.test/is
(and
(= 15 (reduce #'+ [1 2 3 4 5]))
(= 0 (reduce #'+ []))
(= 6 (reduce #'+ 1 [2 3])))))Black Box Testing
;; "Clojure has many sequence types, which act in subtly different ways.
;; The core functions typically convert them into a uniform \"sequence\"
;; type and work with them that way, but it can be important to understand
;; the behavioral and performance differences so that you know which kind
;; is appropriate for your application.<br /><br />Write a function which
;; takes a collection and returns one of :map, :set, :list, or :vector -
;; describing the type of collection it was given.<br />You won't be allowed
;; to inspect their class or use the built-in predicates like list? - the
;; point is to poke at them and understand their behavior.
;;
;; Restrictions (please don't use these function(s)): class, type, Class,
;; vector?, sequential?, list?, seq?, map?, set?, instance?, getClass"
(defn lookup-type [obj]
(let [a [1 1]
result (conj obj a)]
(cond
(and (not (associative? obj)) (= (conj result a) result)) :set
(and (associative? obj) (identical? (conj result a) result)) :map
(and (not (associative? obj)) (identical? (first result) a)) :list
(and (associative? obj) (identical? (last result) a)) :vector
:else (throw (IllegalArgumentException. "Unknown collection type!")))))
(and
(= :map (lookup-type {:a 1, :b 2}))
(= :list (lookup-type (range (rand-int 20))))
(= :vector (lookup-type [1 2 3 4 5 6]))
(= :set (lookup-type #{10 (rand-int 5)}))
(= [:map :set :vector :list] (map lookup-type [{} #{} [] ()])))
Greatest Common Divisor
(defn gcd
"Greatest common dividor of 2 numbers.
See http://en.wikipedia.org/wiki/Greatest_common_divisor"
[a b]
(cond
(or (= 0 a) (= 0 b)) 0
( = a b) a
(> a b) (recur (- a b) b)
:else (recur a (- b a))))
(clojure.test/testing
"Given two integers, write a function which
returns the greatest common divisor."
(clojure.test/is
(and
(= (gcd 2 4) 2)
(= (gcd 10 5) 5)
(= (gcd 5 7) 1)
(= (gcd 1023 858) 33))))Prime Numbers
;; Write a function which returns the first x
;; number of prime numbers.
(defn prime-sieve
"Prime sieve"
([]
(letfn [(add-prime? [candidate primes]
(let [narrowed-primes (reduce-primes-set candidate primes)]
(if (empty? narrowed-primes)
candidate
(recur (next-prime-candidate candidate) primes))))
(reduce-primes-set [candidate primes-set]
(let [max-val (inc (long (Math/ceil (Math/sqrt candidate))))]
(for [i primes-set :while (< i max-val)
:when (zero? (mod candidate i))] i)))
(next-prime-candidate [current-candidate]
(+ 2 current-candidate))
(gen-primes [candidate acc]
(lazy-seq
(let [next-prime (add-prime? candidate acc)]
(cons next-prime
(gen-primes (next-prime-candidate next-prime)
(conj acc next-prime))))))]
(cons 2 (gen-primes 3 [2]))))
([n]
(take n (prime-sieve))))
(and
(= (prime-sieve 2) [2 3])
(= (prime-sieve 5) [2 3 5 7 11])
(= (last (prime-sieve 100)) 541))Recurring Theme
;; Clojure only has one non-stack-consuming looping construct: recur.
;; Either a function or a loop can be used as the recursion point.
;; Either way, recur rebinds the bindings of the recursion point
;; to the values it is passed.
;;
;; Recur must be called from the tail-position,
;; and calling it elsewhere will result in an error.
(= [7 6 5 4 3]
(loop [x 5
result []]
(if (> x 0)
(recur (dec x) (conj result (+ 2 x)))
result)))Merge with a Function
;; Write a function which takes a function f and a variable number of maps.
;; Your function should return a map that consists of the rest of the maps
;; conj-ed onto the first. If a key occurs in more than one map,
;; the mapping(s) from the latter (left-to-right) should be combined
;; with the mapping in the result by calling (f val-in-result val-in-latter)
;;
;; Restrictions (please don't use these function(s)): merge-with
(defn my-merge-with [f m & ms]
(if (empty? ms)
m
(let [new-m (reduce (fn [acc [k v]]
(if (acc k)
(assoc acc k (f (acc k) v))
(assoc acc k v)))
m
(first ms))]
(recur f new-m (rest ms)))))
(and
(= (my-merge-with * {:a 2, :b 3, :c 4} {:a 2} {:b 2} {:c 5})
{:a 4, :b 6, :c 20})
(= (my-merge-with - {1 10, 2 20} {1 3, 2 10, 3 15})
{1 7, 2 10, 3 15})
(= (my-merge-with concat {:a [3], :b [6]} {:a [4 5], :c [8 9]} {:b [7]})
{:a [3 4 5], :b [6 7], :c [8 9]}))Word Sorting
(defn split-sentence [xs]
(->> (re-seq #"\w+|\d+" xs) (sort-by #(.toLowerCase %))))
(clojure.test/testing
"Write a function that splits a sentence up
into a sorted list of words.
Capitalization should not affect sort order
and punctuation should be ignored."
(clojure.test/is
(and
(= (split-sentence "Have a nice day.")
["a" "day" "Have" "nice"])
(= (split-sentence "Clojure is a fun language!")
["a" "Clojure" "fun" "is" "language"])
(= (split-sentence "Fools fall for foolish follies.")
["fall" "follies" "foolish" "Fools" "for"]))))Rearranging Code: ->
;; 4Clojure Question 71
;;
;; The -> macro threads an expression x through a variable
;; number of forms. First, x is inserted as the second item
;; in the first form, making a list of it if it is not a
;; list already.
;; Then the first form is inserted as the second item in
;; the second form, making a list of that form if necessary.
;; This process continues for all the forms.
;; Using -> can sometimes make your code more readable.
;;
(= (last (sort (rest (reverse [2 5 4 1 3 6]))))
(-> [2 5 4 1 3 6] (reverse) (rest) (sort) (last))
5)Rearranging Code: ->>
;; The ->> macro threads an expression x through a variable number of forms.
;; First, x is inserted as the last item in the first form,
;; making a list of it if it is not a list already.
;; Then the first form is inserted as the last item in the second form,
;; making a list of that form if necessary.
;; This process continues for all the forms.
;; Using ->> can sometimes make your code more readable.
(= (reduce + (map inc (take 3 (drop 2 [2 5 4 1 3 6]))))
(->> [2 5 4 1 3 6] (drop 2) (take 3) (map inc) (__))
11)Analyze a Tic-Tac-Toe Board
;; A tic-tac-toe board is represented by a two dimensional vector.
;; X is represented by :x,
;; O is represented by :o,
;; and empty is represented by :e.
;;
;; A player wins by placing three Xs or three Os in a horizontal,
;; vertical, or diagonal row. Write a function which analyzes a
;; tic-tac-toe board and returns :x if X has won, :o if O has won,
;; and nil if neither player has won.
;; Other approach http://mathworld.wolfram.com/MagicSquare.html
;; - Map number to 0 for empty cells
;; - Leave number as is for :o
;; - Multiply the number by 2 for :x
;; - If the total of a row adds up to 15 :o wins, 30 :x wins otherwise nobody
;; (defn tic-tac-toe-winner-magic-square [board]
;; (let [magic-square [[8 1 6] [3 5 7] [4 9 2]]
;; make-groups (fn [board]
;; (let [max-col (count board), max-row (count (first board))]
;; (concat
;; (for [i (range max-col)]
;; (for [j (range max-row)] ((board i) j)))
;; (for [j (range max-row)]
;; (for [i (range max-col)] ((board i) j)))
;; [(for [i (range max-row)] ((board i) i))]
;; [(for [i (reverse (range max-row))]
;; ((board i) (dec (- max-row i))))])))
;; row-winner (fn [row] (case (reduce + row) 15 :o, 30 :x, nil))
;; cell-to-num (fn [cell mapped-cell]
;; (case cell :o mapped-cell, :x (* mapped-cell 2), 0))
;; transform-row (fn [matrix mapped-matrix]
;; (mapv (fn [row mapped-row] (cell-to-num row mapped-row))
;; matrix mapped-matrix))
;; game-winner (fn [cell-groups winner]
;; (if (or winner (empty? cell-groups))
;; winner
;; (recur (next cell-groups) (row-winner (first cell-groups)))))]
;; (let [num-matrix (mapv transform-row board magic-square)
;; cell-groups (make-groups num-matrix)]
;; (game-winner cell-groups nil))))
(defn tic-tac-toe-winner [board]
(letfn [(make-groups [board]
(let [max-col (count board)
max-row (count (first board))]
(concat
(for [i (range max-col)]
(for [j (range max-row)] ((board i) j)))
(for [j (range max-row)]
(for [i (range max-col)] ((board i) j)))
[(for [i (range max-row)] ((board i) i))]
[(for [i (reverse (range max-row))]
((board i) (dec (- max-row i))))])))
(game-winner [[cell-grp & cell-grps]]
(if-not cell-grp
nil
(if (and (not (some #(= :e %1) cell-grp)) (apply = cell-grp))
(first cell-grp)
(recur cell-grps))))]
(game-winner (make-groups board))))
(and
(= nil (tic-tac-toe-winner [[:e :e :e]
[:e :e :e]
[:e :e :e]]))
(= :x (tic-tac-toe-winner [[:x :e :o]
[:x :e :e]
[:x :e :o]]))
(= :o (tic-tac-toe-winner [[:e :x :e]
[:o :o :o]
[:x :e :x]]))
(= nil (tic-tac-toe-winner [[:x :e :o]
[:x :x :e]
[:o :x :o]]))
(= :x (tic-tac-toe-winner [[:x :e :e]
[:o :x :e]
[:o :e :x]]))
(= :o (tic-tac-toe-winner [[:x :e :o]
[:x :o :e]
[:o :e :x]]))
(= nil (tic-tac-toe-winner [[:x :o :x]
[:x :o :x]
[:o :x :o]])))Filter Perfect Squares
;; Perfect square numbers
;; http://en.wikipedia.org/wiki/Square_number
(defn perfect-sqrt-nums [num-seq]
(letfn [(perfect-square? [s]
(let [n-sqrt (Math/sqrt (Integer/valueOf s))]
(= (double 0) (double (- n-sqrt (Math/floor n-sqrt))))))]
(->> (interpose "," (filter perfect-square? (re-seq #"\d+" num-seq)))
(apply str))))
(clojure.test/testing
"Given a string of comma separated integers,
write a function which returns a new comma
separated string that only contains the numbers
which are perfect squares."
(clojure.test/is
(and
(= (perfect-sqrt-nums "4,5,6,7,8,9") "4,9")
(= (perfect-sqrt-nums "15,16,25,36,37") "16,25,36"))))Euler’s Totient Function
;; Write a function which calculates Euler's totient function.
;; NOTE: Reusing gcd function from question 66.
;;
;; Two numbers are coprime if their greatest common divisor equals 1.
;; Euler's totient function f(x) is defined as the number of positive integers
;; less than x which are coprime to x.
;; The special case f(1) equals 1.
;; Write a function which calculates Euler's totient function.
(defn euler-totient [n]
{:pre [ (pos? n)]}
(if (= 1 n) n
(count (filter #(= 1 (gcd n %1)) (range n)))))
(clojure.test/testing
"Test Euler's totient function."
(clojure.test/is
(and
(= (euler-totient 1) 1)
(= (euler-totient 10) (count '(1 3 7 9)) 4)
(= (euler-totient 40) 16)
(= (euler-totient 99) 60))))Intro to Trampoline
;;
;; The trampoline function takes a function f and a variable number of parameters.
;; Trampoline calls f with any parameters that were supplied.
;; If f returns a function, trampoline calls that function with no arguments.
;; This is repeated, until the return value is not a function,
;; and then trampoline returns that non-function value.
;; This is useful for implementing mutually recursive algorithms
;; in a way that won't consume the stack.
(= [1 3 5 7 9 11]
(letfn
[(foo [x y] #(bar (conj x y) y))
(bar [x y] (if (> (last x) 10)
x
#(foo x (+ 2 y))))]
(trampoline foo [] 1)))Anagram Finder
;; 4Clojure Question 77
;;
;; Write a function which finds all the anagrams in a vector of words.
;; A word x is an anagram of word y if all the letters in x can be
;; rearranged in a different order to form y.
;; Your function should return a set of sets,
;; where each sub-set is a group of words which are anagrams of each other.
;; Each sub-set should have at least two words.
;; Words without any anagrams should not be included in the result.
(defn anagrams [xs]
(->> (vals (group-by #(sort %1) (set xs)))
(filter #(> (count %1) 1))
(map set)
(into #{})))
(and
(= (anagrams ["meat" "mat" "team" "mate" "eat"])
#{#{"meat" "team" "mate"}})
(= (anagrams ["veer" "lake" "item" "kale" "mite" "ever"])
#{#{"veer" "ever"} #{"lake" "kale"} #{"mite" "item"}}))Reimplement Trampoline
;; Reimplement the function described in <a href="76"> "Intro to Trampoline"</a>.
;;
;; Restrictions (please don't use these function(s)): trampoline
(defn my-trampoline [f x]
((fn step [f & args]
(let [result (apply f args)]
(if-not (fn? result)
result
(recur f result)))) f x))
(= (letfn [(triple [x] #(sub-two (* 3 x)))
(sub-two [x] #(stop?(- x 2)))
(stop? [x] (if (> x 50) x #(triple x)))]
(__ triple 2))
82)
(= (letfn [(my-even? [x] (if (zero? x) true #(my-odd? (dec x))))
(my-odd? [x] (if (zero? x) false #(my-even? (dec x))))]
(map (partial __ my-even?) (range 6)))
[true false true false true false])Triangle Minimal Path
;; Write a function which calculates the sum of the
;; minimal path through a triangle.
;;
;; The triangle is represented as a collection of vectors.
;; The path should start at the top of the triangle and
;; move to an adjacent number on the next row until the
;; bottom of the triangle is reached.
(defn min-triangle-path [col]
(letfn [(current-row-min-path [cur-row]
(->> (partition 2 1 cur-row) (mapv #(reduce min %))))
(update-triangle-base [last-row min-path]
(mapv + last-row min-path))
(update-triangle [triangle idx updated-base]
(assoc triangle idx updated-base))
(min-path-sum [triangle]
(if (= 1 (count triangle))
(first (flatten triangle))
(let [new-triangle (pop triangle)
new-base (last new-triangle)
new-base-idx (dec (count new-triangle))
prev-base (peek triangle)]
(recur (->> (current-row-min-path prev-base)
(update-triangle-base new-base)
(update-triangle new-triangle new-base-idx))))))]
(min-path-sum (into [] col))))
(and
(= 7 (min-triangle-path '([1]
[2 4]
[5 1 4]
[2 3 4 5]))) ; 1->2->1->3
(= 20 (min-triangle-path '([3]
[2 4]
[1 9 3]
[9 9 2 4]
[4 6 6 7 8]
[5 7 3 5 1 4]))) ; 3->4->3->2->7->1
)Test perfect numbers
;; A number is "perfect" if the sum of its divisors equal the number itself.
;; 6 is a perfect number because 1+2+3=6.
;; Write a function which returns true for perfect numbers and false otherwise.
(defn perfect-num? [n]
(and (not (odd? n))
(= n (reduce + (filter #(= 0 (mod n %)) (range 1 n))))))
(clojure.test/testing
"Test perfect numbers."
(clojure.test/is
(and
(= (perfect-num? 6) true)
(= (perfect-num? 7) false)
(= (perfect-num? 496) true)
(= (perfect-num? 500) false)
(= (perfect-num? 8128) true))))Write a function which returns the intersection of two sets.
;; The intersection is the sub-set of items that each set has in common.
;; Restrictions (please don't use these function(s)): intersection
(defn set-intersection [x1 x2]
(set (filter x1 x2)))
(clojure.test/testing
"Intersection of two sets."
(clojure.test/is
(and
(= (set-intersection #{0 1 2 3} #{2 3 4 5}) #{2 3})
(= (set-intersection #{0 1 2} #{3 4 5}) #{})
(= (set-intersection #{:a :b :c :d} #{:c :e :a :f :d}) #{:a :c :d}))))Test continuous word chain
;; A word chain consists of a set of words ordered so that each word
;; differs by only one letter from the words directly before and after it.
;; The one letter difference can be either an insertion, a deletion,
;; or a substitution.
;; Here is an example word chain:
;; cat -> cot -> coat -> oat -> hat -> hot -> hog -> dog
;; Write a function which takes a sequence of words,
;; and returns true if they can be arranged into one continous word chain,
;; and false if they cannot.
(defn continuous-word-chain?
[xs]
(letfn [(combinations-tree [elem xs]
(cons elem
(when-not (empty? xs)
(let [r (filter #(= 1 (words-diff elem %1)) xs)]
(when-not (empty? r)
(map #(combinations-tree %1 (disj xs %1)) r))))))
(max-tree-height [tree]
(if (not (seq? tree)) 0
(+ 1 (reduce max (map max-tree-height tree)))))
(words-diff [w1 w2]
(let [r (count (apply disj (set w1) (seq w2)))
letters-diff (- (count w1) (count w2))
diff (if (neg? letters-diff) (* -1 letters-diff) letters-diff)]
(+ diff r)))]
(->> (map #(max-tree-height (combinations-tree %1 (disj xs %1))) xs)
(reduce max)
(= (count xs)))))
(clojure.test/testing
"Word chain"
(clojure.test/is
(and
(= true (continuous-word-chain? #{"hat" "coat" "dog" "cat"
"oat" "cot" "hot" "hog"}))
(= false (continuous-word-chain? #{"cot" "hot" "bat" "fat"}))
(= false (continuous-word-chain? #{"to" "top" "stop" "tops" "toss"}))
(= true (continuous-word-chain? #{"spout" "do" "pot" "pout" "spot" "dot"}))
(= true (continuous-word-chain? #{"share" "hares" "shares" "hare" "are"}))
(= false (continuous-word-chain? #{"share" "hares" "hare" "are"})))))A Half-Truth
;; Write a function which takes a variable number of booleans.
;; Your function should return true if some of the parameters
;; are true, but not all of the parameters are true.
;; Otherwise your function should return false.
(defn some-true? [& cols]
(= (set cols) #{true false}))
(and (= false (some-true? false false))
(= true (some-true? true false))
(= false (some-true? true))
(= true (some-true? false true false))
(= false (some-true? true true true))
(= true (some-true? true true true false)))Transitive Closure
;; http://en.wikipedia.org/wiki/Transitive_closure
;; http://en.wikipedia.org/wiki/Binary_relation
;;
;; Write a function which generates the transitive closure of a binary relation.
;; The relation will be represented as a set of 2 item vectors.
(defn transitive-closure [g]
(letfn [(adjacent-vertices [v g]
(keep (fn [[a b]] (when (= v a) b)) g))
(vertices [g]
(set (reduce concat g)))
(bfs-iter [v g]
(loop [queue (-> (clojure.lang.PersistentQueue/EMPTY) (conj v))
visited #{v}]
(if (empty? queue)
(disj visited v)
(let [node (peek queue)
cur-queue (pop queue)
neighbours (adjacent-vertices node g)
unseen (reduce disj (set neighbours) visited)
next-queue (reduce conj cur-queue unseen)]
(recur next-queue (reduce conj visited unseen))))))
(transitive-edges [v connections]
(reduce #(conj %1 [v %2]) connections))]
(reduce (fn [graph vertex]
(let [reachable-nodes (bfs-iter vertex g)
transitive-links (transitive-edges vertex reachable-nodes)]
(reduce #(conj %1 %2) graph transitive-links)))
g (vertices g))))
(and
(let [divides #{[8 4] [9 3] [4 2] [27 9]}]
(= (transitive-closure divides) #{[4 2] [8 4] [8 2] [9 3] [27 9] [27 3]}))
(let [more-legs
#{["cat" "man"] ["man" "snake"] ["spider" "cat"]}]
(= (transitive-closure more-legs)
#{["cat" "man"] ["cat" "snake"] ["man" "snake"]
["spider" "cat"] ["spider" "man"] ["spider" "snake"]}))
(let [progeny
#{["father" "son"] ["uncle" "cousin"] ["son" "grandson"]}]
(= (transitive-closure progeny)
#{["father" "son"] ["father" "grandson"]
["uncle" "cousin"] ["son" "grandson"]})))Power Set
;; http://en.wikipedia.org/wiki/Power_set
;; Write a function which generates the power set of a given set.
;;
;; The power set of a set x is the set of all subsets of x,
;; including the empty set and x itself.
;;
;; http://www.mathsisfun.com/sets/power-set.html
(defn powerset [xs]
(let [ln (count xs)
col (into [] xs)
set-size (Math/pow 2 ln)]
(->> (for [i (range set-size)]
(->> (for [j (range ln)
:when (pos? (bit-and i (bit-shift-left 1 j)))]
(col j))
(into #{})))
(into #{}))))
(and
(= (powerset #{1 :a}) #{#{1 :a} #{:a} #{} #{1}})
(= (powerset #{}) #{#{}})
(= (powerset #{1 2 3})
#{#{} #{1} #{2} #{3} #{1 2} #{1 3} #{2 3} #{1 2 3}})
(= (count (powerset (into #{} (range 10)))) 1024))
Test happy numbers
;; Happy numbers are positive integers that follow a particular formula:
;; - Take each individual digit, square it,
;; and then sum the squares to get a new number.
;; - Repeat with the new number and eventually,
;; you might get to a number whose squared sum is 1.
;; - This is a happy number.
;; An unhappy number (or sad number) is one that loops endlessly.
;; Write a function that determines if a number is happy or not.
(defn happy-num? [n]
{:pre [(pos? n)]}
(letfn [(digits [n] (map #(Character/getNumericValue %1) (str n)))
(square-sum [xs] (long (reduce #(+ %1 (Math/pow %2 2)) 0 xs)))]
(loop [loop-detection #{}
i n]
(let [sum (square-sum (digits i))]
(cond
(= 1 sum) true
(contains? loop-detection sum) false
:else (recur (conj loop-detection sum) sum))))))
(and
(= (happy-num? 7) true)
(= (happy-num? 986543210) true)
(= (happy-num? 2) false)
(= (happy-num? 3) false))THERE IS NO PROBLEM 87
Symmetric difference of two sets
;; Write a function which returns the symmetric difference of two sets.
;; The symmetric difference is the set of items belonging to one
;; but not both of the two sets.
(defn symetric-set-diff [s1 s2]
(let [not-in-s1 (filter #(not (s1 %1)) s2)
not-in-s2 (filter #(not (s2 %1)) s1)]
(set (concat not-in-s1 not-in-s2))))
(and
(= (symetric-set-diff #{1 2 3 4 5 6} #{1 3 5 7}) #{2 4 6 7})
(= (symetric-set-diff #{:a :b :c} #{}) #{:a :b :c})
(= (symetric-set-diff #{} #{4 5 6}) #{4 5 6})
(= (symetric-set-diff #{[1 2] [2 3]} #{[2 3] [3 4]}) #{[1 2] [3 4]}))Graph Tour
;; Starting with a graph you must write a function that returns true
;; if it is possible to make a tour of the graph in which every edge
;; is visited exactly once.The graph is represented by a
;; vector of tuples, where each tuple represents a single edge.
;;
;; The rules are:
;; - You can start at any node.
;; - You must visit each edge exactly once.
;; - All edges are undirected.
(defn eulerian-walk? [g]
(letfn [(vertices [g] (set (reduce concat g)))
(adjacent-edges [v g] (filter (fn [[a b]] (or (= a v) (= b v))) g))
(next-vertex [e u] (first (disj (into #{} e) u)))
(rem-first [xs x]
(when xs
(if (not= (first xs) x)
(cons (first xs) (rem-first (next xs) x))
(rest xs))))
(get-paths [v g q]
(let [edges (adjacent-edges v g)]
(if (empty? edges)
(cons q nil)
(->> (mapcat #(reduce conj [] (get-paths (next-vertex %1 v)
(rem-first g %1)
(conj q %1)))
edges)
(map seq)))))]
(let [nodes (vertices g)
node (first nodes)
eulerian-trail? (->> (get-paths node g (clojure.lang.PersistentQueue/EMPTY))
(some #(= (count g) (count %1))))]
(or eulerian-trail? false))))
(and (= true (eulerian-walk? [[:a :b]]))
(= false (eulerian-walk? [[:a :a] [:b :b]]))
(= false (eulerian-walk? [[:a :b] [:a :b] [:a :c] [:c :a]
[:a :d] [:b :d] [:c :d]]))
(= true (eulerian-walk? [[1 2] [2 3] [3 4] [4 1]]))
(= true (eulerian-walk? [[:a :b]
[:a :c]
[:c :b]
[:a :e]
[:b :e]
[:a :d]
[:b :d]
[:c :e]
[:d :e]
[:c :f]
[:d :f]]))
(= false (eulerian-walk? [[1 2] [2 3] [2 4] [2 5]])))Cartesian product
;; Write a function which calculates the Cartesian product of two sets.
;; http://en.wikipedia.org/wiki/Cartesian_product
(defn cartesian-product [xs1 xs2]
(set (for [x1 xs1 x2 xs2] [x1 x2])))
(and
(= (cartesian-product #{"ace" "king" "queen"} #{"♠" "♥" "♦" "♣"})
#{["ace" "♠"] ["ace" "♥"] ["ace" "♦"] ["ace" "♣"]
["king" "♠"] ["king" "♥"] ["king" "♦"] ["king" "♣"]
["queen" "♠"] ["queen" "♥"] ["queen" "♦"] ["queen" "♣"]})
(= (cartesian-product #{1 2 3} #{4 5})
#{[1 4] [2 4] [3 4] [1 5] [2 5] [3 5]})
(= 300 (count (cartesian-product (into #{} (range 10))
(into #{} (range 30))))))
Check if a graph is connected
;; Given a graph, determine whether the graph is connected.
;; A connected graph is such that a path exists between any two given nodes.
;; - Your function must return true if the graph is connected and false otherwise.
;; - You will be given a set of tuples representing the edges of a graph.
;; - Each member of a tuple being a vertex/node in the graph.
;; - Each edge is undirected (can be traversed either direction).
(defn graph-connected? [g]
(letfn [(adjacent-nodes [v g]
(keep (fn [[a b]] (cond (= v a) b, (= v b) a)) g))
(dfs-iter [v g]
(loop [stack (cons v nil) visited #{}]
(if (empty? stack)
visited
(let [node (peek stack)
col (pop stack)
not-visited? (not (contains? visited node))
next-visited (if not-visited?
(conj visited node)
visited)
next-stack (if not-visited?
(reduce #(conj %1 %2)
col (adjacent-nodes node g))
col)]
(recur next-stack next-visited)))))]
(let [nodes (set (reduce concat g))
connections (reduce (fn [acc v]
(assoc acc v (dfs-iter v g)))
{} nodes)]
(every? #(= (count nodes) (count (last %1))) connections))))
(and
(= true (graph-connected? #{[:a :a]}))
(= true (graph-connected? #{[:a :b]}))
(= false (graph-connected? #{[1 2] [2 3] [3 1]
[4 5] [5 6] [6 4]}))
(= true (graph-connected? #{[1 2] [2 3] [3 1]
[4 5] [5 6] [6 4] [3 4]}))
(= false (graph-connected? #{[:a :b] [:b :c] [:c :d]
[:x :y] [:d :a] [:b :e]}))
(= true (graph-connected? #{[:a :b] [:b :c] [:c :d]
[:x :y] [:d :a] [:b :e] [:x :a]})))
Roman numerals to decimal parser Also read about the substractive principle on Wikipedia.
;; Roman numerals are easy to recognize,
;; but not everyone knows all the rules necessary to work with them.
;; Write a function to parse a Roman-numeral string and return the number it represents.
;;
;; You can assume that the input will be well-formed, in upper-case,
;; and follow the subtractive principle.
;;
;; You don't need to handle any numbers greater than MMMCMXCIX (3999),
;; the largest number representable with ordinary letters.
(defn roman-numeral-to-number [str]
(let [sym-table {\I 1, \V 5, \X 10, \L 50, \C 100, \D 500, \M 1000}
nums (mapv #(sym-table %1) str)]
(reduce + (map-indexed (fn [idx item]
(let [max-right-item (reduce max (subvec nums idx))
num-x (if (> max-right-item item) -1 1)]
(* item num-x)))
nums))))
(and (= 14 (roman-numeral-to-number "XIV"))
(= 827 (roman-numeral-to-number "DCCCXXVII"))
(= 3999 (roman-numeral-to-number "MMMCMXCIX"))
(= 48 (roman-numeral-to-number "XLVIII")))
Partially Flatten a Sequence
;; Write a function which flattens any nested combination of sequential things
;; (lists, vectors, etc.), but maintains the lowest level sequential items.
;; The result should be a sequence of sequences with only one level of nesting.
;; bottom up approach seems easier here for me at least here...
(defn flatten-1 [xs]
(letfn [(stepper [xs]
(when (not (empty? xs))
(let [cur (last xs)]
(if-not (and (coll? cur) (every? coll? cur))
(cons cur (stepper (butlast xs)))
(recur (concat (butlast xs) cur))))))]
(into '() (stepper xs))))
(and (= (flatten-1 [["Do"] ["Nothing"]])
[["Do"] ["Nothing"]])
(= (flatten-1 [[[[:a :b]]] [[:c :d]] [:e :f]])
[[:a :b] [:c :d] [:e :f]])
(= (flatten-1 '((1 2)((3 4)((((5 6)))))))
'((1 2)(3 4)(5 6))))
Game of Life
;; The <a href="http://en.wikipedia.org/wiki/Conway's_Game_of_Life">game of life</a>
;; is a cellular automaton devised by mathematician John Conway.
;; The 'board' consists of both live (#) and dead ( ) cells.
;; Each cell interacts with its eight neighbours (horizontal, vertical, diagonal),
;; and its next state is dependent on the following rules:<br/><br/>
;; 1) Any live cell with fewer than two live neighbours dies, as if caused by
;; under-population.
;; 2) Any live cell with two or three live neighbours lives on to the next generation.
;; 3) Any live cell with more than three live neighbours dies, as if by overcrowding.
;; 4) Any dead cell with exactly three live neighbours becomes a live cell,
;; as if by reproduction.
;; Write a function that accepts a board, and returns a board representing
;; the next generation of cells.
(defn game-of-life [board]
(letfn [(find-cells [board]
(map-indexed (fn [i row]
(map-indexed (fn [j ch]
(let [ln (count (live-neighbors board [i j]))]
(if (= ch \#)
(if (or (< ln 2) (> ln 3)) \space ch)
(if (= ln 3) \# ch))))
row))
board))
(valid-live-cell? [board [i j] max-i max-j]
(and (>= i 0) (>= j 0) (< i max-i) (< j max-j) (= (get-in board [i j]) \#)))
(live-neighbors [board [i j]]
(let [max-i (count board)
max-j (count (first board))]
(->> [[(dec i) j] [(dec i) (dec j)] [i (dec j)] [(inc i) (dec j)]
[(inc i) j] [(inc i) (inc j)] [i (inc j)] [(dec i) (inc j)]]
(filter #(valid-live-cell? board %1 max-i max-j)))))]
(mapv #(apply str %1) (find-cells board))))
(and
(= (game-of-life [" "
" ## "
" ## "
" ## "
" ## "
" "])
[" "
" ## "
" # "
" # "
" ## "
" "])
(= (game-of-life [" "
" "
" ### "
" "
" "])
[" "
" # "
" # "
" # "
" "])
(= (game-of-life [" "
" "
" ### "
" ### "
" "
" "])
[" "
" # "
" # # "
" # # "
" # "
" "]))To Tree, or not to Tree
;; 4Clojure Question 95
;;
;; Write a predicate which checks whether or not a given
;; sequence represents a
;; <a href="http://en.wikipedia.org/wiki/Binary_tree">binary tree</a>.
;; Each node in the tree must have a value, a left child, and a right child.
(defn binary-tree? [xs]
(letfn [(valid-node? [col idx]
(let [elem (nth col idx)]
(if (nil? elem)
true
(and (coll? elem) (binary-tree? elem)))))]
(if (= 3 (count xs))
(and
(not (nil? (nth xs 0))) (valid-node? xs 1) (valid-node? xs 2))
false)))
(and
(= (binary-tree? '(:a (:b nil nil) nil))
true)
(= (binary-tree? '(:a (:b nil nil)))
false)
(= (binary-tree? [1 nil [2 [3 nil nil] [4 nil nil]]])
true)
(= (binary-tree? [1 [2 nil nil] [3 nil nil] [4 nil nil]])
false)
(= (binary-tree? [1 [2 [3 [4 nil nil] nil] nil] nil])
true)
(= (binary-tree? [1 [2 [3 [4 false nil] nil] nil] nil])
false)
(= (binary-tree? '(:a nil ()))
false))Beauty is Symmetry
;; Let us define a binary tree as "symmetric" if the left
;; half of the tree is the mirror image of the right half
;; of the tree.
;; Write a predicate to determine whether or not a given
;; binary tree is symmetric. (see <a href='/problem/95'>To Tree,
;; or not to Tree</a> for a reminder on the tree representation we're using).
;; Reusing our binary-tree? function from problem 95.
(defn symmetric-binary-tree? [xs]
(letfn [(symmetric? [left right]
(if (or (coll? left) (coll? right))
(and (and (coll? left) (coll? right))
(= (first left) (first right))
(symmetric? (second (rest left)) (first (rest right)))
(symmetric? (first (rest left)) (second (rest right))))
(= left right)))]
(and (binary-tree? xs) (symmetric? (first (rest xs)) (second (rest xs))))))
(and
(= (symmetric-binary-tree? '(:a (:b nil nil) (:b nil nil))) true)
(= (symmetric-binary-tree? '(:a (:b nil nil) nil)) false)
(= (symmetric-binary-tree? '(:a (:b nil nil) (:c nil nil))) false)
(= (symmetric-binary-tree? [1 [2 nil [3 [4 [5 nil nil] [6 nil nil]] nil]]
[2 [3 nil [4 [6 nil nil] [5 nil nil]]] nil]])
true)
(= (symmetric-binary-tree? [1 [2 nil [3 [4 [5 nil nil] [6 nil nil]] nil]]
[2 [3 nil [4 [5 nil nil] [6 nil nil]]] nil]])
false)
(= (symmetric-binary-tree? [1 [2 nil [3 [4 [5 nil nil] [6 nil nil]] nil]]
[2 [3 nil [4 [6 nil nil] nil]] nil]])
false))Pascal’s Triangle
;; <a href="http://en.wikipedia.org/wiki/Pascal%27s_triangle">Pascal's triangle</a>
;; is a triangle of numbers computed using the following rules:<br/></br>-
;; The first row is 1.</br>- Each successive row is computed by adding
;; together adjacent numbers in the row above, and adding a 1 to the
;; beginning and end of the row.<br/><br/>Write a function which
;; returns the nth row of Pascal's Triangle.
(defn pascal-triangle-row [n]
(letfn [(step [xs]
(-> (into [(first xs)] (map #(reduce +' %1N) (partition 2 1 xs)))
(conj (last xs))))]
(last (take n (iterate step [1])))))
(and
(= (pascal-triangle-row 1) [1])
(= (map pascal-triangle-row (range 1 6))
[ [1]
[1 1]
[1 2 1]
[1 3 3 1]
[1 4 6 4 1]])
(= (pascal-triangle-row 11)
[1 10 45 120 210 252 210 120 45 10 1]))Equivalence Classes
;; A function f defined on a domain D induces an
;; <a href="http://en.wikipedia.org/wiki/Equivalence_relation">equivalence relation</a>
;; on D, as follows: a is equivalent to b with respect to f if and only if (f a)
;; is equal to (f b).
;; Write a function with arguments f and D that computes the
;; <a href="http://en.wikipedia.org/wiki/Equivalence_class">equivalence classes</a>
;; of D with respect to f.
(defn equiv-classes [f xs]
(->> (vals (group-by f xs))
(map set)
(into #{})))
(and
(= (equiv-classes #(* % %) #{-2 -1 0 1 2})
#{#{0} #{1 -1} #{2 -2}})
(= (equiv-classes #(rem % 3) #{0 1 2 3 4 5 })
#{#{0 3} #{1 4} #{2 5}})
(= (equiv-classes identity #{0 1 2 3 4})
#{#{0} #{1} #{2} #{3} #{4}})
(= (equiv-classes (constantly true) #{0 1 2 3 4})
#{#{0 1 2 3 4}}))
Product Digits
;; Write a function which multiplies two numbers
;; and returns the result as a sequence of its digits.
(defn product-digits [a b]
(map #(Character/getNumericValue %) (str (* a b))))
(and
(= (product-digits 1 1) [1])
(= (product-digits 99 9) [8 9 1])
(= (product-digits 999 99) [9 8 9 0 1]))Least Common Multiple
;; Write a function which calculates the
;; <a href="http://en.wikipedia.org/wiki/Least_common_multiple">least common multiple</a>.
;; Your function should accept a variable number of positive integers or ratios.
;; We reuse our gcd function from question 66
(defn lcm [& nums]
(reduce (fn [a b] (/ (* a b) (gcd a b))) nums))
(and
(== (lcm 2 3) 6)
(== (lcm 5 3 7) 105)
(== (lcm 1/3 2/5) 2)
(== (lcm 3/4 1/6) 3/2)
(== (lcm 7 5/7 2 3/5) 210))Levenshtein Distance
;; https://secure.wikimedia.org/wikipedia/en/wiki/Levenshtein_distance
;; Given two sequences x and y, calculate the Levenshtein distance of x
;; and y, i. e. the minimum number of edits needed to transform x into y.
;; The allowed edits are:<br/><br/>- insert a single item<br/>-
;; delete a single item<br/>- replace a single item with another item
;; <br/><br/>WARNING: Some of the test cases may timeout
;; if you write an inefficient solution!
(defn levenshtein [w1 w2]
(letfn [(cell-value [same-char? prev-row cur-row col-idx]
(min (inc (nth prev-row col-idx))
(inc (last cur-row))
(+ (nth prev-row (dec col-idx)) (if same-char? 0 1))))]
(loop [row-idx 1
max-rows (inc (count w2))
prev-row (range (inc (count w1)))]
(if (= row-idx max-rows)
(last prev-row)
(let [ch2 (nth w2 (dec row-idx))
next-prev (reduce (fn [cur-row i]
(let [same-char? (= (nth w1 (dec i)) ch2)]
(conj cur-row (cell-value same-char?
prev-row
cur-row
i))))
[row-idx]
(range 1 (count prev-row)))]
(recur (inc row-idx) max-rows, next-prev))))))
(and
(= (levenshtein "kitten" "sitting") 3)
(= (levenshtein "closure" "clojure") (levenshtein "clojure" "closure") 1)
(= (levenshtein "xyx" "xyyyx") 2)
(= (levenshtein "" "123456") 6)
(= (levenshtein "Clojure" "Clojure") (levenshtein "" "") (levenshtein [] []) 0)
(= (levenshtein [1 2 3 4] [0 2 3 4 5]) 2)
(= (levenshtein '(:a :b :c :d) '(:a :d)) 2)
(= (levenshtein "ttttattttctg" "tcaaccctaccat") 10)
(= (levenshtein "gaattctaatctc" "caaacaaaaaattt") 9))intoCamelCase
;; When working with java, you often need to create an object
;; with <code>fieldsLikeThis</code>, but you'd rather work with a
;; hashmap that has <code>:keys-like-this</code> until it's time to convert.
;; Write a function which takes lower-case hyphen-separated strings and
;; converts them to camel-case strings.
(defn camel-case [s]
(let [->camel-case #(concat (str (Character/toUpperCase (first %1))) (rest %1))
xs (take-nth 2 (partition-by #(= \- %1) s))]
(->> (mapcat ->camel-case (rest xs)) (concat (first xs)) (apply str))))
(and
(= (camel-case "something") "something")
(= (camel-case "multi-word-key") "multiWordKey")
(= (camel-case "leaveMeAlone") "leaveMeAlone"))Generating k-combinations
;; Given a sequence S consisting of n elements generate all
;; <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Combination">k-combinations</a>
;; of S, i. e. generate all possible sets consisting of k distinct elements taken from S.
;;
;; The number of k-combinations for a sequence is equal to the
;; <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Binomial_coefficient">
;; binomial coefficient</a>.
;; Reusing our powerset function from problem 85 and we just reduce it.
(defn k-distinct-combos [n xs]
(case (- n (count xs))
pos? #{}
zero? #{xs}
(into #{} (filter #(= n (count %1)) (powerset xs)))))
(and
(= (k-distinct-combos 1 #{4 5 6}) #{#{4} #{5} #{6}})
(= (k-distinct-combos 10 #{4 5 6}) #{})
(= (k-distinct-combos 2 #{0 1 2}) #{#{0 1} #{0 2} #{1 2}})
(= (k-distinct-combos 3 #{0 1 2 3 4}) #{#{0 1 2} #{0 1 3} #{0 1 4} #{0 2 3} #{0 2 4}
#{0 3 4} #{1 2 3} #{1 2 4} #{1 3 4} #{2 3 4}})
(= (k-distinct-combos 4 #{[1 2 3] :a "abc" "efg"}) #{#{[1 2 3] :a "abc" "efg"}})
(= (k-distinct-combos 2 #{[1 2 3] :a "abc" "efg"}) #{#{[1 2 3] :a} #{[1 2 3] "abc"}
#{[1 2 3] "efg"} #{:a "abc"}
#{:a "efg"} #{"abc" "efg"}}))Write roman numerals
;; This is the inverse of Problem 92, but much easier.
;; Given an integer smaller than 4000, return the corresponding roman numeral
;; in uppercase, adhering to the subtractive principle.
;; The trick here is to select carefully
;; the symbol table so that you don't need to check for repetition
;; 4 = IV vs XXXX (without storing such cases,
;; it is more difficult for the recursion...)
;; I couldn't make it work with a basic table as in problem 92.
(defn number-to-roman-numeral [n]
(let [sym-table {1 "I", 4 "IV", 5 "V", 9 "IX",
10 "X", 40 "XL", 50 "L", 90 "XC",
100 "C", 400 "CD", 500 "D", 900 "CM"
1000 "M"}
sym-keys (keys sym-table)]
(loop [remainder n
result []]
(if (zero? remainder)
(apply str result)
(let [min-num-sym (reduce max (filter #(<= %1 remainder) sym-keys))
min-sym (sym-table min-num-sym)]
(recur (- remainder min-num-sym) (conj result min-sym)))))))
(and
(= "I" (number-to-roman-numeral 1))
(= "XXX" (number-to-roman-numeral 30))
(= "IV" (number-to-roman-numeral 4))
(= "CXL" (number-to-roman-numeral 140))
(= "DCCCXXVII" (number-to-roman-numeral 827))
(= "MMMCMXCIX" (number-to-roman-numeral 3999))
(= "XLVIII" (number-to-roman-numeral 48)));; Given an input sequence of keywords and numbers,
;; create a map such that each key in the map is a keyword,
;; and the value is a sequence of all the numbers (if any)
;; between it and the next keyword in the sequence.
(defn keyword-set[xs]
(->> (reduce (fn [acc item]
(if (keyword? item)
(conj acc item [])
(assoc acc (dec (count acc)) (conj (peek acc) item))))
[]
xs)
(apply hash-map)))
(and
(= {} (keyword-set []))
(= {:a [1]} (keyword-set [:a 1]))
(= {:a [1], :b [2]} (keyword-set [:a 1, :b 2]))
(= {:a [1 2 3], :b [], :c [4]} (keyword-set [:a 1 2 3 :b :c 4])))
Number Maze
;; Given a pair of numbers, the start and end point,
;; find a path between the two using only three possible operations:<ul>
;;
;; <li>double</li>
;; <li>halve (odd numbers cannot be halved)</li>
;; <li>add 2</li></ul>
;;
;; Find the shortest path through the "maze".
;; Because there are multiple shortest paths, you must return the
;; length of the shortest path, not the path itself.
(defn num-maze-steps [start end]
(letfn [(possible-operations [n]
(let [ops [#(+ %1 2), #(* %1 2)]]
(if (even? n)
(conj ops #(/ %1 2))
ops)))
(apply-operations [nums]
(mapcat (fn [n] (map #(%1 n) (possible-operations n))) nums))]
(loop [nodes [start]
step 1]
(if (some #(= end %) nodes)
step
(recur (apply-operations nodes), (inc step))))))
(and
(= 1 (num-maze-steps 1 1)) ; 1
(= 3 (num-maze-steps 3 12)) ; 3 6 12
(= 3 (num-maze-steps 12 3)) ; 12 6 3
(= 3 (num-maze-steps 5 9)) ; 5 7 9
(= 9 (num-maze-steps 9 2)) ; 9 18 20 10 12 6 8 4 2
(= 5 (num-maze-steps 9 12)) ; 9 11 22 24 12
)Simple closures
;; <p>Lexical scope and first-class functions are two of the most
;; basic building blocks of a functional language like Clojure.
;; When you combine the two together, you get something very
;; powerful called <strong>lexical closures</strong>.
;;
;; With these, you can exercise a great deal of control over the
;; lifetime of your local bindings, saving their values for use later,
;; long after the code you're running now has finished.</p>
;;
;;
;; <p>It can be hard to follow in the abstract, so let's build a
;; simple closure. Given a positive integer <i>n</i>, return a
;;'' function <code>(f x)</code> which computes <i>x<sup>n</sup></i>.
;; Observe that the effect of this is to preserve the value of
;; <i>n</i> for use outside the scope in which it is defined.</p>
(defn exp [n]
(fn [& args]
(long (Math/pow (first args) n))))
(and (= 256 ((exp 2) 16),
((exp 8) 2))
(= [1 8 27 64] (map (exp 3) [1 2 3 4]))
(= [1 2 4 8 16] (map #((exp %) 2) [0 1 2 3 4])))Lazy Searching
;; 4Clojure Question 108
;;
;; <p>Given any number of sequences, each sorted from smallest to largest,
;; find the smallest single number which appears in all of the sequences.
;; The sequences may be infinite, so be careful to search lazily.</p>
(defn smallest-common-num [& cols]
(letfn [(move-cursor [col num]
(if (>= (first col) num)
col
(recur (rest col) num)))]
(let [firsts (map first cols)
max-first (reduce max firsts)]
(if (apply = firsts)
(first firsts)
(recur (map #(move-cursor %1 max-first) cols))))))
(and
(= 3 (smallest-common-num [3 4 5]))
(= 4 (smallest-common-num [1 2 3 4 5 6 7] [0.5 3/2 4 19]))
(= 7 (smallest-common-num (range) (range 0 100 7/6) [2 3 5 7 11 13]))
(= 64 (smallest-common-num (map #(* % % %) (range)) ;; perfect cubes
(filter #(zero? (bit-and % (dec %))) (range)) ;; powers of 2
(iterate inc 20))) ;; at least as large as 20
)THERE IS NO PROBLEM 109
;; <p>Write a function that returns a lazy sequence of "pronunciations"
;; of a sequence of numbers. A pronunciation of each element in the
;; sequence consists of the number of repeating identical numbers
;; and the number itself. For example, <code>[1 1]</code> is
;; pronounced as <code>[2 1]</code> ("two ones"), which in turn is
;; pronounced as <code>[1 2 1 1]</code> ("one two, one one").</p>
;; <p>Your function should accept an initial sequence of numbers,
;; and return an infinite lazy sequence of pronunciations,
;; each element being a pronunciation of the previous element.</p>
(defn pronunciations [xs]
(letfn [(stepper [xs]
(mapcat #(vector (count %1) (first %1)) (partition-by identity xs)))]
(iterate stepper (stepper xs))))
(and
(= [[1 1] [2 1] [1 2 1 1]] (take 3 (pronunciations [1])))
(= [3 1 2 4] (first (pronunciations [1 1 1 4 4])))
(= [1 1 1 3 2 1 3 2 1 1] (nth (pronunciations [1]) 6))
(= 338 (count (nth (pronunciations [3 2]) 15))));; Write a function that takes a string and a partially-filled
;; crossword puzzle board, and determines if the input string
;; can be legally placed onto the board.
;; The crossword puzzle board consists of a collection of
;; partially-filled rows. Empty spaces are denoted with an
;; underscore (_), unusable spaces are denoted with a hash
;; symbol(#), and pre-filled spaces have a character in place;
;; the whitespace characters are for legibility and should be ignored.
;;
;; For a word to be legally placed on the board:
;; - It may use empty spaces (underscores)
;; - It may use but must not conflict with any pre-filled characters.
;; - It must not use any unusable spaces (hashes).;;
;; - There must be no empty spaces (underscores) or extra characters
;; before or after the word (the word may be bound by unusable spaces though).
;; - Characters are not case-sensitive.
;; - Words may be placed vertically (proceeding top-down only),
;; or horizontally (proceeding left-right only).
;; Rewrite using plain stupid brute force instead of smarter code.
;;
;; 0. always go down, check right and down, switch column as needed .
;; 1. If the placement is valid (nothing other than # before and after), just go to 2.
;; 2. Replace non spaces or overwritable chars by #.
;; 3. If we have a range without # we have a solution.
(defn solved-crossword? [word board]
(let [wc (count word)
puzzle (mapv (fn [col] (into [] (filter #(not= \space %1) col))) board)
max-i (count puzzle)
max-j (count (first puzzle))]
(letfn [(solved? [puzzle i j]
(if (= \# ((puzzle i) j))
false
(or
;; right
(and (>= (- max-j j wc) 0)
(valid-position? puzzle i (dec j))
(valid-position? puzzle i (+ j wc))
(not (some #(= \# %1)
(transform (subvec (puzzle i)
j
(+ j wc))))))
;; down
(and (>= (- max-i i wc) 0)
(valid-position? puzzle (dec i) j)
(valid-position? puzzle (+ i wc) j)
(not (some #(= \# %1)
(transform (subvec (mapv (fn [m] (nth m j)) puzzle)
i
(+ i wc)))))))))
(valid-position? [puzzle i j]
(if (or (neg? i) (neg? j) (>= j max-j) (>= i max-i))
true
(= \# ((puzzle i) j))))
(transform [view]
(map-indexed #(if (and (not= %2 \_) (not= (nth word %1) %2)) \# %2) view))
(next-move [puzzle i j]
(let [next-i (mod (inc i) max-i)
next-j (if (>= (inc i) max-i) (inc j) j)]
[next-i next-j]))
(solve-puzzle [puzzle [i j] cur-iter max-iter]
(if (solved? puzzle i j)
true
(if (= cur-iter max-iter)
false
(recur puzzle (next-move puzzle i j) (inc cur-iter) max-iter))))]
(solve-puzzle puzzle [0 0] 1 (* max-i max-j)))))
(and (= true (solved-crossword? "the" ["_ # _ _ e"]))
(= false (solved-crossword? "the" ["c _ _ _"
"d _ # e"
"r y _ _"]))
(= true (solved-crossword? "joy" ["c _ _ _"
"d _ # e"
"r y _ _"]))
(= false (solved-crossword? "joy" ["c o n j"
"_ _ y _"
"r _ _ #"]))
(= true (solved-crossword? "clojure" ["_ _ _ # j o y"
"_ _ o _ _ _ _"
"_ _ f _ # _ _"])))Sequs Horribilis
;; Create a function which takes an integer
;; and a nested collection of integers as arguments.
;;
;; Analyze the elements of the input collection and
;; return a sequence which maintains the nested
;; structure, and which includes all elements starting
;; from the head whose sum is less than or equal to
;; the input integer.
(defn sequs-horribilis [n xs]
(letfn [(mk-ctx [q cur sum limit]
(if (coll? cur)
(let [pending (enqueue [] cur sum limit)
new-sum (+ sum (reduce + (flatten pending)))]
{:queue (conj q pending) :sum new-sum})
{:queue (maybe-add-num q cur sum limit) :sum (+ sum cur)}))
(maybe-add-num [q cur sum limit]
(if (> (+ cur sum) limit) q (conj q cur)))
(enqueue [q xs sum limit]
(if (or (empty? xs) (> sum limit))
q
(let [cur (first xs)
ctx (mk-ctx q cur sum limit)]
(recur (:queue ctx) (next xs) (:sum ctx) limit))))]
(enqueue [] xs 0 n)))
(and
(= (sequs-horribilis 10 [1 2 [3 [4 5] 6] 7])
'(1 2 (3 (4))))
(= (sequs-horribilis 30 [1 2 [3 [4 [5 [6 [7 8]] 9]] 10] 11])
'(1 2 (3 (4 (5 (6 (7)))))))
(= (sequs-horribilis 9 (range))
'(0 1 2 3))
(= (sequs-horribilis 1 [[[[[1]]]]])
'(((((1))))))
(= (sequs-horribilis 0 [1 2 [3 [4 5] 6] 7])
'())
(= (sequs-horribilis 0 [0 0 [0 [0]]])
'(0 0 (0 (0))))
(= (sequs-horribilis 1 [-10 [1 [2 3 [4 5 [6 7 [8]]]]]])
'(-10 (1 (2 3 (4))))))Data types
;; Write a function that takes a variable number of integer arguments.
;; If the output is coerced into a string, it should return a comma
;; (and space) separated list of the inputs sorted smallest to largest.
;; If the output is coerced into a sequence, it should return a seq of
;; unique input elements in the same order as they were entered.
;;
;; Restrictions (please don't use these function(s)): proxy
(defn seqable-proxy [& xs]
(let [input (apply list xs)]
(reify clojure.lang.Seqable
(toString [this]
(clojure.string/join ", " (sort input)))
(seq [this]
(seq (distinct input))))))
(and (= "1, 2, 3" (str (seqable-proxy 2 1 3)))
(= '(2 1 3) (seq (seqable-proxy 2 1 3)))
(= '(2 1 3) (seq (seqable-proxy 2 1 3 3 1 2)))
(= '(1) (seq (apply seqable-proxy (repeat 5 1))))
(= "1, 1, 1, 1, 1" (str (apply seqable-proxy (repeat 5 1))))
(and (= nil (seq (seqable-proxy)))
(= "" (str (seqable-proxy)))))Global take-while
;; take-while is great for filtering sequences,
;; but it limited: you can only examine
;; a single item of the sequence at a time. What if you need to keep
;; track of some state as you go over the sequence?
;;
;; Write a function which accepts an integer <code>n</code>,
;; a predicate <code>p</code>, and a sequence. It should return
;; a lazy sequence of items in the list up to, but not including,
;; the <code>n</code>th item that satisfies the predicate.
;;
(defn take-up-to [n pred? xs]
(lazy-seq
(when (and (pos? n) (not (empty? xs)))
(let [cur (first xs)
next-n (if (pred? cur)
(dec n)
n)]
(when-not (zero? next-n)
(cons cur (take-up-to next-n pred? (rest xs))))))))
(and (= [2 3 5 7 11 13]
(take-up-to 4 #(= 2 (mod % 3))
[2 3 5 7 11 13 17 19 23]))
(= ["this" "is" "a" "sentence"]
(take-up-to 3 #(some #{\i} %)
["this" "is" "a" "sentence" "i" "wrote"]))
(= ["this" "is"]
(take-up-to 1 #{"a"}
["this" "is" "a" "sentence" "i" "wrote"])))The Balance of N
;; Problem 115
;;
;; A balanced number is one whose component digits
;; have the same sum on the left and right halves of the number.
;;
;; Write a function which accepts an integer n,
;; and returns true iff n is balanced.
(defn balanced-num? [n]
(let [digits (mapv #(Character/getNumericValue %1) (str n))
mid (fn [xs] (let [middle (bigint (/ (count xs) 2))]
(if (odd? (count xs))
[middle (inc middle)]
[middle middle])))
bounds (mid digits)]
(= (reduce +' (subvec digits 0 (first bounds)))
(reduce +' (subvec digits (last bounds))))))
(and
(= true (balanced-num? 11))
(= true (balanced-num? 121))
(= false (balanced-num? 123))
(= true (balanced-num? 0))
(= false (balanced-num? 88099))
(= true (balanced-num? 89098))
(= true (balanced-num? 89089))
(= (take 20 (filter balanced-num? (range)))
[0 1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 101]))Prime Sandwich
;; http://en.wikipedia.org/wiki/Balanced_prime
;;
;; A balanced prime is a prime number which is also
;; the mean of the primes directly before and after
;; it in the sequence of valid primes.
;; Create a function which takes an integer n,
;; and returns true iff it is a balanced prime.
;;
;; requires previous prime-sieve function from problem 67
(defn balanced-prime? [n]
(letfn [(primes-interval [n]
(let [[a b c & more] (prime-sieve)]
(loop [acc [a b c] primes more]
(if (> (last acc) n)
acc
(recur (conj (vec (rest acc)) (first primes))
(rest primes))))))]
(let [ [a b c] (primes-interval n)]
(and (= n b) (= n (/ (+ a c) 2))))))
(and (= false (balanced-prime? 4))
(= true (balanced-prime? 563))
(= 1103 (nth (filter balanced-prime? (range)) 15)))For Science!
;; A mad scientist with tenure has created an experiment
;; tracking mice in a maze. Several mazes have been
;; randomly generated, and you've been tasked with
;; writing a program to determine the mazes in which
;; it's possible for the mouse to reach the cheesy endpoint.
;;
;; Write a function which accepts a maze in the form of a
;; collection of rows, each row is a string where:
;;
;; <ul>
;;
;; <li>spaces represent areas where the mouse can walk freely</li>
;;
;; <li>hashes (#) represent walls where the mouse can not walk</li>
;;
;; <li>M represents the mouse's starting point</li>
;;
;; <li>C represents the cheese which the mouse must reach</li>
;;
;; </ul>
;;
;; The mouse is not allowed to travel diagonally in the maze
;; (only up/down/left/right), nor can he escape the edge
;; of the maze. Your function must return true iff
;; the maze is solvable by the mouse.
;; Notes:
;; Use graph concepts explicitly for training purposes
;; 1. Make a labelled graph of coordinates and keep only navigable paths.
;; 2. Find the mouse position.
;; 3. Jump to adjacent edges (coordinates) and check the label.
;; 4. Could have written 10-15 less lines of code without formalities.
(defn cheese-reachable? [maze]
(letfn [(unsigned-int [n] (if (>= n 0) n (* n -1)))
(make-labelled-graph [maze]
(->> (for [i (range (count maze))]
(for [j (range (count (first maze)))
:let [label (get-in maze [i j])]
:when (not= label \#)]
[i j label]))
(reduce concat)
set))
(next-edges [[u v _] g]
(filter (fn [[a b _]]
(and (or (= a u) (= b v))
(= 1 (unsigned-int (- (or (first (reduce disj
(set [u v])
[a b]))
u)
(or (first (reduce disj
(set [a b])
[u v]))
a))))))
g))
(bfs-iter-match? [maze]
(let [graph (make-labelled-graph maze)
start (first (filter (fn [[i j label]] (= label \M)) graph))]
(loop [visits (-> (clojure.lang.PersistentQueue/EMPTY) (conj start))
seen #{start}]
(if (empty? visits)
false
(let [cur (peek visits)
cur-visits (pop visits)]
(if (= (last cur) \C)
true
(let [next-steps (->> (next-edges cur (disj graph cur))
(filter #(not (contains? seen %1))))
next-visits (reduce conj cur-visits next-steps)
next-seen (reduce conj seen next-steps)]
(recur next-visits next-seen))))))))]
(bfs-iter-match? maze)))
(and
(= true (cheese-reachable? ["M C"]))
(= false (cheese-reachable? ["M # C"]))
(= true (cheese-reachable? ["#######"
"# #"
"# # #"
"#M # C#"
"#######"]))
(= false (cheese-reachable? ["########"
"#M # #"
"# # #"
"# # # #"
"# # #"
"# # #"
"# # # #"
"# # #"
"# # C#"
"########"]))
(= false (cheese-reachable? ["M "
" "
" "
" "
" ##"
" #C"]))
(= true (cheese-reachable? ["C######"
" # "
" # # "
" # #M"
" # "]))
(= true (cheese-reachable? ["C# # # #"
" "
"# # # # "
" "
" # # # #"
" "
"# # # #M"])))Re-implement Map
;; Problem 118
;;
;; <p>Map is one of the core elements of a functional programming language.
;; Given a function <code>f</code> and an input sequence <code>s</code>,
;; return a lazy sequence of <code>(f x)</code> for each element
;; <code>x</code> in <code>s</code>.
;;
;; Restrictions (please don't use these function(s)): map, map-indexed, mapcat, for
(defn do-map [f col]
((fn step [xs]
(lazy-seq
(when-not (empty? xs)
(cons (f (first xs)) (step (rest xs))))))
col))
(and (= [3 4 5 6 7]
(do-map inc [2 3 4 5 6]))
(= (repeat 10 nil)
(do-map (fn [_] nil) (range 10)))
(= [1000000 1000001]
(->> (do-map inc (range))
(drop (dec 1000000))
(take 2))))Win at Tic-Tac-Toe
;; <p>As in <a href="/problem/73">Problem 73</a>, a tic-tac-toe board
;; is represented by a two dimensional vector.
;; X is represented by :x, O is represented by :o,
;; and empty is represented by :e.
;; Create a function that accepts a game piece and board as arguments,
;; and returns a set (possibly empty) of all valid board placements
;; of the game piece which would result in an immediate win.</p>
;; <p>Board coordinates should be as in calls to <code>get-in</code>.
;; For example, <code>[0 1]</code> is the topmost row, center position.</p>
(defn win-tic-tac-toe-moves [piece board]
(letfn [(winning-moves [piece board cell-groups]
(reduce (fn [acc cell-group]
(let [group-vals (mapv (fn [[i j]] ((board i) j)) cell-group)
empty-idxs (keep-indexed #(when (= %2 :e) %1) group-vals)]
(if (and (= 1 (count empty-idxs))
(= 2 (count (filter #(= piece %1) group-vals))))
(conj acc (nth cell-group (first empty-idxs)))
acc)))
#{} cell-groups))
(make-groups [board]
(let [max-col (count board)
max-row (count (first board))]
(concat
(for [i (range max-col)] (for [j (range max-row)] [i j]))
(for [j (range max-row)] (for [i (range max-col)] [i j]))
[(for [i (range max-row)] [i i])]
[(for [i (reverse (range max-row))] [i (dec (- max-row i))])])))]
(->> (make-groups board) (winning-moves piece board))))
(and
(= (win-tic-tac-toe-moves :x
[[:o :e :e]
[:o :x :o]
[:x :x :e]])
#{[2 2] [0 1] [0 2]})
(= (win-tic-tac-toe-moves :x [[:x :o :o]
[:x :x :e]
[:e :o :e]])
#{[2 2] [1 2] [2 0]})
(= (win-tic-tac-toe-moves :x [[:x :e :x]
[:o :x :o]
[:e :o :e]])
#{[2 2] [0 1] [2 0]})
(= (win-tic-tac-toe-moves :x [[:x :x :o]
[:e :e :e]
[:e :e :e]])
#{})
(= (win-tic-tac-toe-moves :o [[:x :x :o]
[:o :e :o]
[:x :e :e]])
#{[2 2] [1 1]}))Sum of square of digits
;; 4Clojure Question 120
;;
;; Write a function which takes a collection of integers
;; as an argument. Return the count of how many elements
;; are smaller than the sum of their squared component
;; digits.
;;
;; For example: 10 is larger than 1 squared plus 0 squared;
;; whereas 15 is smaller than 1 squared plus 5 squared.
(defn cnt-<-x2-sum-digits [xs]
(letfn [(digits [n]
(lazy-seq
(loop [x n r '()]
(if (< x 10) (cons x r)
(recur (quot x 10) (cons (mod x 10) r))))))
(square-sum [col]
(reduce + (map (fn [x] (Math/pow x 2)) col)))]
(reduce (fn [total num]
(let [num-digits (digits num)
num-square-sum (square-sum num-digits)]
(if (< num num-square-sum) (inc total) total)))
0
xs)))
(and (= 8 (cnt-<-x2-sum-digits (range 10)))
(= 19 (cnt-<-x2-sum-digits (range 30)))
(= 50 (cnt-<-x2-sum-digits (range 100)))
(= 50 (cnt-<-x2-sum-digits (range 1000))))Universal Computation Engine
;; Given a mathematical formula in prefix notation, return a function that calculates
;; the value of the formula.
;; The formula can contain nested calculations using the four basic
;; mathematical operators, numeric constants, and symbols representing variables.
;; The returned function has to accept a single parameter containing the map
;; of variable names to their values.
(defn uce [expr]
(let [fn-mappings (apply hash-map ['+ + '- - '* * '/ /])]
(letfn [(map-args [st mappings]
(map (fn [sym]
(if (coll? sym)
(map-args sym mappings)
(get mappings sym sym)))
st))
(visit [[op & arguments]]
(if (nil? arguments)
(recur op)
(if-not (some coll? arguments)
(apply (fn-mappings op) arguments)
(visit
(cons op (map (fn [x]
(if (coll? x)
(visit x)
x))
arguments))))))]
(fn [& args]
(->> (map-args expr (first args)) visit)))))
(and
(= 2 ((uce '(/ a b))
'{b 8 a 16}))
(= 8 ((uce '(+ a b 2))
'{a 2 b 4}))
(= [6 0 -4]
(map (uce '(* (+ 2 a)
(- 10 b)))
'[{a 1 b 8}
{b 5 a -2}
{a 2 b 11}]))
(= 1 ((uce '(/ (+ x 2)
(* 3 (+ y 1))))
'{x 4 y 1})))Read a binary number
;; Convert a binary number, provided in the form of a string, to its numerical value.
(defn bin->decimal [xs]
(let [col (map #(read-string %) (reverse (re-seq #"\d" xs)))]
(bigint (reduce +' (map-indexed #(* (Math/pow 2 %1) %2) col)))))
(defn bin->decimal-java [str-xs]
(Integer/valueOf str-xs 2))
(and (= 0 (bin->decimal "0"))
(= 7 (bin->decimal "111"))
(= 8 (bin->decimal "1000"))
(= 9 (bin->decimal "1001"))
(= 255 (bin->decimal "11111111"))
(= 1365 (bin->decimal "10101010101"))
(= 65535 (bin->decimal "1111111111111111")))
THERE IS NO PROBLEM 123
Analyze Reversi-TODO
;; <p><a href="http://en.wikipedia.org/wiki/Reversi">Reversi</a>
;; is normally played on an 8 by 8 board.
;; In this problem, a 4 by 4 board is represented as a two-dimensional
;; vector with black, white, and empty pieces represented by 'b, 'w, and 'e,
;; respectively.
;; Create a function that accepts a game board and color as arguments,
;; and returns a map of legal moves for that color.
;; Each key should be the coordinates of a legal move,
;; and its value a set of the coordinates of the pieces flipped by that move.
;; <p>Board coordinates should be as in calls to get-in.
;; For example, <code>[0 1]</code> is the topmost row, second column from the left.
(defn reversi
[board color])
(and
(= {[1 3] #{[1 2]}, [0 2] #{[1 2]}, [3 1] #{[2 1]}, [2 0] #{[2 1]}}
(reversi '[[e e e e]
[e w b e]
[e b w e]
[e e e e]] 'w))
(= {[3 2] #{[2 2]}, [3 0] #{[2 1]}, [1 0] #{[1 1]}}
(reversi '[[e e e e]
[e w b e]
[w w w e]
[e e e e]] 'b))
(= {[0 3] #{[1 2]}, [1 3] #{[1 2]}, [3 3] #{[2 2]}, [2 3] #{[2 2]}}
(reversi '[[e e e e]
[e w b e]
[w w b e]
[e e b e]] 'w))
(= {[0 3] #{[2 1] [1 2]}, [1 3] #{[1 2]}, [2 3] #{[2 1] [2 2]}}
(reversi '[[e e w e]
[b b w e]
[b w w e]
[b w w w]] 'b)))
Gus’ Quinundrum-TODO
;; Create a function of no arguments which returns a string that
;; is an <i>exact</i> copy of the function itself.
;; Hint: read <a href="http://en.wikipedia.org/wiki/Quine_(computing)">this</a>
;; if you get stuck (this question is harder than it first appears);
;; but it's worth the effort to solve it independently if you can!
;; Fun fact: Gus is the name of the <a href="http://i.imgur.com/FBd8z.png">4Clojure dragon</a>.
(= (str '__) (__))Through the Looking Class
;; Enter a value which satisfies the following:
(let [x Class]
(and (= (class x) x) x))Love Triangle-TODO
;; Everyone loves triangles, and it's easy to understand why—
;; they're so wonderfully symmetric (except scalenes, they suck).
;; Your passion for triangles has led you to become a miner
;; (and part-time Clojure programmer) where you work all day to
;; chip out isosceles-shaped minerals from rocks gathered in a
;; nearby open-pit mine.
;; There are too many rocks coming from the mine to harvest them
;; all so you've been tasked with writing a program to analyze
;; the mineral patterns of each rock, and determine which rocks
;; have the biggest minerals.
;; Someone has already written a
;; <a href="http://en.wikipedia.org/wiki/Computer_vision">computer-vision</a>
;; system for the mine.
;; It images each rock as it comes into the processing centre and
;; creates a cross-sectional
;; <a href="http://en.wikipedia.org/wiki/Bit_array">bitmap</a>
;; of mineral (1) and rock (0) concentrations for each one.
;; You must now create a function which accepts a collection of integers,
;; each integer when read in base-2 gives the bit-representation of the
;; rock (again, 1s are mineral and 0s are worthless scalene-like rock).
;; You must return the cross-sectional area of the largest harvestable
;; mineral from the input rock, as follows:
;; <li>The minerals only have smooth faces when sheared vertically
;; or horizontally from the rock's cross-section</li>
;;
;; <li>The mine is only concerned with harvesting isosceles triangles
;; (such that one or two sides can be sheared)</li>
;;
;; <li>If only one face of the mineral is sheared, its opposing vertex
;; must be a point (ie. the smooth face must be of odd length), and
;; its two equal-length sides must intersect the shear face at 45°
;; (ie. those sides must cut even-diagonally)</li>
;;
;; <li>The harvested mineral may not contain any traces of rock</li>
;;
;; <li>The mineral may lie in any orientation in the plane</li>
;;
;; <li>Area should be calculated as the sum of 1s that comprise the mineral</li>
;;
;; <li>Minerals must have a minimum of three measures of area to be harvested</li>
;;
;; <li>If no minerals can be harvested from the rock, your function should return nil</li>
;;
;; </ul>
(defn love-triangle [xs]
(let [mk-digits (fn [xs]
(->> (map #(Integer/toBinaryString %1) xs)
(mapv (fn [x]
(mapv #(Character/getNumericValue %1) x)))))
get-triangles (fn [tree]
)
]
(mk-digits xs)
))
(and
(= 10 (love-triangle [15 15 15 15 15]))
; 1111 1111
; 1111 *111
; 1111 -> **11
; 1111 ***1
; 1111 ****
(= 15 (love-triangle [1 3 7 15 31]))
; 00001 0000*
; 00011 000**
; 00111 -> 00***
; 01111 0****
; 11111 *****
(= 3 (love-triangle [3 3]))
; 11 *1
; 11 -> **
(= 4 (love-triangle [7 3]))
; 111 ***
; 011 -> 0*1
(= 6 (love-triangle [17 22 6 14 22]))
; 10001 10001
; 10110 101*0
; 00110 -> 00**0
; 01110 0***0
; 10110 10110
(= 9 (love-triangle [18 7 14 14 6 3]))
; 10010 10010
; 00111 001*0
; 01110 01**0
; 01110 -> 0***0
; 00110 00**0
; 00011 000*1
(= nil (love-triangle [21 10 21 10]))
; 10101 10101
; 01010 01010
; 10101 -> 10101
; 01010 01010
(= nil (love-triangle [0 31 0 31 0]))
; 00000 00000
; 11111 11111
; 00000 -> 00000
; 11111 11111
; 00000 00000
)Recognize Playing Cards
;; A standard American deck of playing cards has four suits
;; - spades, hearts, diamonds, and clubs - and thirteen cards in each suit.
;; Two is the lowest rank, followed by other integers up to ten;
;; then the jack, queen, king, and ace.
;; It's convenient for humans to represent these cards as suit/rank pairs,
;; such as H5 or DQ: the heart five and diamond queen respectively.
;; But these forms are not convenient for programmers,
;; so to write a card game you need some way to parse an input string
;; into meaningful components. For purposes of determining rank,
;; we will define the cards to be valued from 0 (the two) to 12 (the ace).
;; Write a function which converts (for example) the string "SJ" into a map
;; of <code>{:suit :spade, :rank 9}</code>. A ten will always be represented
;; with the single character "T", rather than the two characters "10".
(defn decode-card [code]
(let [suits {\D :diamond \H :heart \S :spade \C :club}
ranks {\A 12 \K 11 \Q 10 \T 10 \J 9}
get-rank (fn [x]
(cond
(Character/isDigit x) (- (Character/getNumericValue x) 2)
(= \T x) (- (ranks x) 2)
:else (ranks x)))]
(-> (assoc {} :suit (suits (first code)))
(assoc :rank (get-rank (last code))))))
(and
(= {:suit :diamond :rank 10} (decode-card "DQ"))
(= {:suit :heart :rank 3} (decode-card "H5"))
(= {:suit :club :rank 12} (decode-card "CA"))
(= (range 13) (map (comp :rank decode-card str)
'[S2 S3 S4 S5 S6 S7
S8 S9 ST SJ SQ SK SA])))THERE IS NO PROBLEM 129
Tree reparenting-TODO
;; 4Clojure Question 130
;;
;; Every node of a tree is connected to each of its children as
;;
;; well as its parent. One can imagine grabbing one node of
;;
;; a tree and dragging it up to the root position, leaving all
;;
;; connections intact. For example, below on the left is
;;
;; a binary tree. By pulling the "c" node up to the root, we
;;
;; obtain the tree on the right.
;;
;; <br/>
;;
;; <img src="http://i.imgur.com/UtD2T.png">
;;
;; <br/>
;;
;; Note it is no longer binary as "c" had three connections
;;
;; total -- two children and one parent.
;;
;;
;;
;; Each node is represented as a vector, which always has at
;;
;; least one element giving the name of the node as a symbol.
;;
;; Subsequent items in the vector represent the children of the
;;
;; node. Because the children are ordered it's important that
;;
;; the tree you return keeps the children of each node in order
;;
;; and that the old parent node, if any, is appended on the
;;
;; right.
;;
;;
;;
;; Your function will be given two args -- the name of the node
;;
;; that should become the new root, and the tree to transform.
;;
;;
;;
;; Use M-x 4clojure-check-answers when you're done!
(= '(n)
(__ 'n '(n)))
(= '(a (t (e)))
(__ 'a '(t (e) (a))))
(= '(e (t (a)))
(__ 'e '(a (t (e)))))
(= '(a (b (c)))
(__ 'a '(c (b (a)))))
(= '(d
(b
(c)
(e)
(a
(f
(g)
(h)))))
(__ 'd '(a
(b
(c)
(d)
(e))
(f
(g)
(h)))))
(= '(c
(d)
(e)
(b
(f
(g)
(h))
(a
(i
(j
(k)
(l))
(m
(n)
(o))))))
(__ 'c '(a
(b
(c
(d)
(e))
(f
(g)
(h)))
(i
(j
(k)
(l))
(m
(n)
(o))))))
Sum Some Set Subsets
;; Given a variable number of sets of integers,
;; create a function which returns true iff all of
;; the sets have a non-empty subset with an equivalent summation.
;; Reusing powerset function from Question 85
(defn common-subset-sum? [& xs]
(letfn [(non-empty-subsets [col]
(keep #(not-empty %1) (powerset col)))
(subsets-sum [col]
(into #{} (map #(reduce + %1) col)))
(find-common-subsets [subsets-sums]
(reduce #(filter (set %1) %2) (first subsets-sums) (rest subsets-sums)))]
(if (= 1 (count xs))
true
(->> (map non-empty-subsets xs) (map subsets-sum) find-common-subsets count pos?))))
(and
(= true (common-subset-sum? #{-1 1 99}
#{-2 2 888}
#{-3 3 7777}))
; ex. all sets have a
; subset which sums to zero
(= false (common-subset-sum? #{1}
#{2}
#{3}
#{4}))
(= true (common-subset-sum? #{1}))
(= false (common-subset-sum? #{1 -3 51 9}
#{0}
#{9 2 81 33}))
(= true (common-subset-sum? #{1 3 5}
#{9 11 4}
#{-3 12 3}
#{-3 4 -2 10}))
(= false (common-subset-sum? #{-1 -2 -3 -4 -5 -6}
#{1 2 3 4 5 6 7 8 9}))
(= true (common-subset-sum? #{1 3 5 7}
#{2 4 6 8}))
(= true (common-subset-sum? #{-1 3 -5 7 -9 11 -13 15}
#{1 -3 5 -7 9 -11 13 -15}
#{1 -1 2 -2 4 -4 8 -8}))
(= true (common-subset-sum? #{-10 9 -8 7 -6 5 -4 3 -2 1}
#{10 -9 8 -7 6 -5 4 -3 2 -1})))Insert between two items
;; Write a function that takes a two-argument predicate,
;; a value, and a collection; and returns a new collection
;; where the <code>value</code> is inserted between every
;; two items that satisfy the predicate.
(defn insert-between-items [pred sep xs]
(let [col (partition-all 2 1 xs)]
((fn step [items]
(lazy-seq
(when-not (empty? items)
(let [[prev cur] (first items)]
(if cur
(if (pred prev cur)
(cons prev (cons sep (step (rest items))))
(cons prev (step (rest items))))
(cons prev nil))))))
col)))
(and
(= '(1 :less 6 :less 7 4 3) (insert-between-items < :less [1 6 7 4 3]))
(= '(2) (insert-between-items > :more [2]))
(= [0 1 :x 2 :x 3 :x 4] (insert-between-items #(and (pos? %) (< % %2)) :x (range 5)))
(empty? (insert-between-items > :more ()))
(and (= [0 1 :same 1 2 3 :same 5 8 13 :same 21]
(take 12 (->> [0 1]
(iterate (fn [[a b]] [b (+ a b)]))
(map first) ; fibonacci numbers
(insert-between-items (fn [a b] ; both even or both odd
(= (mod a 2) (mod b 2)))
:same))))))
THERE IS NO PROBLEM 133
A nil key
;; Write a function which, given a key and map,
;; returns true if the map contains an entry with that key and its value is nil.
(defn nil-key? [k m]
(nil? (m k false)))
(and (true? (nil-key? :a {:a nil :b 2}))
(false? (nil-key? :b {:a nil :b 2}))
(false? (nil-key? :c {:a nil :b 2})))Infix Calculator
;; Your friend Joe is always whining about Lisps
;; using the prefix notation for math.
;; Show him how you could easily write a function that does math
;; using the infix notation. Is your favorite language that flexible, Joe?
;;
;; Write a function that accepts a variable length mathematical
;; expression consisting of numbers and the operations +, -, *,
;; and /. Assume a simple calculator that does not do precedence
;; and instead just calculates left to right.
(defn infix-calc
[& expr]
((fn step [init [a f b & more]]
(if-not f
init
(let [r (f a b)]
(recur r (cons r more)))))
0 expr))
(and (= 7 (infix-calc 2 + 5))
(= 42 (infix-calc 38 + 48 - 2 / 2))
(= 8 (infix-calc 10 / 2 - 1 * 2))
(= 72 (infix-calc 20 / 2 + 2 + 4 + 8 - 6 - 10 * 9)))THERE IS NO PROBLEM 136
Digits and bases
;; Write a function which returns a sequence of digits
;; of a non-negative number (first argument) in numerical
;; system with an arbitrary base (second argument).
;; Digits should be represented with their integer values,
;; e.g. 15 would be [1 5] in base 10, [1 1 1 1] in base 2
;; and [15] in base 16.
(defn seq-digits [num radix]
((fn step [r n base]
(if (pos? n)
(let [d (mod n base)
q (quot n base)]
(recur (cons d r) q base))
(if (empty? r) '(0) r)))
'() num radix))
(and (= [1 2 3 4 5 0 1] (seq-digits 1234501 10))
(= [0] (seq-digits 0 11))
(= [1 0 0 1] (seq-digits 9 2))
(= [1 0] (let [n (rand-int 100000)](seq-digits n n)))
(= [16 18 5 24 15 1] (seq-digits Integer/MAX_VALUE 42)))Squares Squared-TODO
;; Create a function of two integer arguments: the start and end,
;; respectively.
;; You must create a vector of strings which renders a 45° rotated square
;; of integers which are successive squares from the start point up to and
;; including the end point.
;; If a number comprises multiple digits, wrap them around the shape individually.
;; If there are not enough digits to complete the shape, fill in the rest with
;; asterisk characters.
;; The direction of the drawing should be clockwise, starting from the center
;; of the shape and working outwards, with the initial direction being down
;; and to the right.
(defn squares-squared [start end]
(let [build-seq (fn [start end]
(loop [i start j end acc []]
(if (> i j)
acc
(let [next-i (Math/pow i 2)]
(recur next-i j (conj acc (long i)))))))
rotate-45 (fn [xs] xs)
make-matrix (fn [nums]
(mapv #(str %1) nums))]
(->> (build-seq start end)
(make-matrix)
(rotate-45))))
(and
(= (squares-squared 2 2) ["2"])
(= (squares-squared 2 4) [" 2 "
"* 4"
" * "])
(= (squares-squared 3 81) [" 3 "
"1 9"
" 8 "])
(= (squares-squared 4 20) [" 4 "
"* 1"
" 6 "])
(= (squares-squared 2 256) [" 6 "
" 5 * "
"2 2 *"
" 6 4 "
" 1 "])
(= (squares-squared 10 10000) [" 0 "
" 1 0 "
" 0 1 0 "
"* 0 0 0"
" * 1 * "
" * * "
" * "]))THERE IS NO PROBLEM 139
Veitch, Please!-TODO
;; Create a function which accepts as input a boolean algebra function
;; in the form of a set of sets, where the inner sets are collections
;; of symbols corresponding to the input boolean variables which
;; satisfy the function (the inputs of the inner sets are conjoint,
;; and the sets themselves are disjoint... also known as canonical minterms).
;; Note: capitalized symbols represent truth, and lower-case symbols
;; represent negation of the inputs.
;; Your function must return the minimal function which is logically
;; equivalent to the input.
;;
;; You may want to give this a read before proceeding:
;; <a href="http://en.wikipedia.org/wiki/K_map">K-Maps</a>
(= (__ #{#{'a 'B 'C 'd}
#{'A 'b 'c 'd}
#{'A 'b 'c 'D}
#{'A 'b 'C 'd}
#{'A 'b 'C 'D}
#{'A 'B 'c 'd}
#{'A 'B 'c 'D}
#{'A 'B 'C 'd}})
#{#{'A 'c}
#{'A 'b}
#{'B 'C 'd}})
(= (__ #{#{'A 'B 'C 'D}
#{'A 'B 'C 'd}})
#{#{'A 'B 'C}})
(= (__ #{#{'a 'b 'c 'd}
#{'a 'B 'c 'd}
#{'a 'b 'c 'D}
#{'a 'B 'c 'D}
#{'A 'B 'C 'd}
#{'A 'B 'C 'D}
#{'A 'b 'C 'd}
#{'A 'b 'C 'D}})
#{#{'a 'c}
#{'A 'C}})
(= (__ #{#{'a 'b 'c}
#{'a 'B 'c}
#{'a 'b 'C}
#{'a 'B 'C}})
#{#{'a}})
(= (__ #{#{'a 'B 'c 'd}
#{'A 'B 'c 'D}
#{'A 'b 'C 'D}
#{'a 'b 'c 'D}
#{'a 'B 'C 'D}
#{'A 'B 'C 'd}})
#{#{'a 'B 'c 'd}
#{'A 'B 'c 'D}
#{'A 'b 'C 'D}
#{'a 'b 'c 'D}
#{'a 'B 'C 'D}
#{'A 'B 'C 'd}})
(= (__ #{#{'a 'b 'c 'd}
#{'a 'B 'c 'd}
#{'A 'B 'c 'd}
#{'a 'b 'c 'D}
#{'a 'B 'c 'D}
#{'A 'B 'c 'D}})
#{#{'a 'c}
#{'B 'c}})
(= (__ #{#{'a 'B 'c 'd}
#{'A 'B 'c 'd}
#{'a 'b 'c 'D}
#{'a 'b 'C 'D}
#{'A 'b 'c 'D}
#{'A 'b 'C 'D}
#{'a 'B 'C 'd}
#{'A 'B 'C 'd}})
#{#{'B 'd}
#{'b 'D}})
(= (__ #{#{'a 'b 'c 'd}
#{'A 'b 'c 'd}
#{'a 'B 'c 'D}
#{'A 'B 'c 'D}
#{'a 'B 'C 'D}
#{'A 'B 'C 'D}
#{'a 'b 'C 'd}
#{'A 'b 'C 'd}})
#{#{'B 'D}
#{'b 'd}})Tricky card games
;; In <a href="http://en.wikipedia.org/wiki/Trick-taking_game">trick-taking
;; card games</a> such as bridge, spades, or hearts, cards are played
;; in groups known as "tricks" - each player plays a single card, in
;; order; the first player is said to "lead" to the trick. After all
;; players have played, one card is said to have "won" the trick. How
;; the winner is determined will vary by game, but generally the winner
;; is the highest card played <i>in the suit that was
;; led</i>.
;; Sometimes (again varying by game), a particular suit will
;; be designated "trump", meaning that its cards are more powerful than
;; any others: if there is a trump suit, and any trumps are played,
;; then the highest trump wins regardless of what was led.
;; Your goal is to devise a function that can determine which of a
;; number of cards has won a trick. You should accept a trump suit, and
;; return a function <code>winner</code>. Winner will be called on a
;; sequence of cards, and should return the one which wins the
;; trick.
;; Cards will be represented in the format returned
;; by <a href="/problem/128/">Problem 128, Recognize Playing Cards</a>:
;; a hash-map of <code>:suit</code> and a
;; numeric <code>:rank</code>. Cards with a larger rank are stronger.
(defn tricky-card-game-winner [trump-suit]
(fn [args]
(let [effective-trump-suit (or trump-suit (:suit (first args)))]
(->> (filter #(= (:suit %1) effective-trump-suit) args)
(apply max-key :rank)))))
(and
(let [notrump (tricky-card-game-winner nil)]
(and (= {:suit :club :rank 9} (notrump [{:suit :club :rank 4}
{:suit :club :rank 9}]))
(= {:suit :spade :rank 2} (notrump [{:suit :spade :rank 2}
{:suit :club :rank 10}]))))
(= {:suit :club :rank 10} ((tricky-card-game-winner :club) [{:suit :spade :rank 2}
{:suit :club :rank 10}]))
(= {:suit :heart :rank 8}
((tricky-card-game-winner :heart) [{:suit :heart :rank 6}
{:suit :heart :rank 8}
{:suit :diamond :rank 10}
{:suit :heart :rank 4}])))THERE IS NO PROBLEM 142
Dot Product
;; Problem 143
;; http://en.wikipedia.org/wiki/Dot_product#Definition
;;
;; Create a function that computes the dot product of two sequences.
;; You may assume that the vectors will have the same length.
(defn dot-product
"dot product of two sequences"
[c1 c2]
{:pre [(= (count c1) (count c2))]}
(reduce + (map * c1 c2)))
(and (= 0 (dot-product [0 1 0] [1 0 0]))
(= 3 (dot-product [1 1 1] [1 1 1]))
(= 32 (dot-product [1 2 3] [4 5 6]))
(= 256 (dot-product [2 5 6] [100 10 1])))Oscillate iterate
;; Problem 144
;;
;; Write an oscillating iterate: a function that takes an initial
;; value and a variable number of functions. It should return a
;; lazy sequence of the functions applied to the value in order,
;; restarting from the first function after it hits the end.
;; Did not know details about 'reductions' function
;; (reductions f init col)
(defn oscillating-iterate
[n & [f fs]]
(cons n
(lazy-seq
(when f
(oscillating-iterate (f n) fs)))))
(and (= (take 4 (oscillating-iterate 3.14 int double)) [3.14 3 3.0])
(= (take 5 (oscillating-iterate 3 #(- % 3) #(+ 5 %))) [3 0 5 2 7])
(= (take 12 (oscillating-iterate 0 inc dec inc dec inc)) [0 1 0 1 0 1 2 1 2 1 2 3]))
For the win
;; Clojure's for macro is a tremendously versatile mechanism
;; for producing a sequence based on some other sequence(s).
;; It can take some time to understand how to use it properly,
;; but that investment will be paid back with clear,
;; concise sequence-wrangling later.
;;
;; With that in mind, read over these <code>for</code> expressions
;; and try to see how each of them produces the same result.
;;
;; answer (take-nth 4 (range 1 40))
(and (= (take-nth 4 (range 1 40)) (for [x (range 40)
:when (= 1 (rem x 4))]
x))
(= (take-nth 4 (range 1 40)) (for [x (iterate #(+ 4 %) 0)
:let [z (inc x)]
:while (< z 40)]
z))
(= (take-nth 4 (range 1 40)) (for [[x y] (partition 2 (range 20))]
(+ x y))))Trees into tables
;; <p>Because Clojure's <code>for</code> macro allows you to "walk"
;; over multiple sequences in a nested fashion, it is excellent for
;; transforming all sorts of sequences. If you don't want a sequence
;; as your final output (say you want a map), you are often still
;; best-off using <code>for</code>, because you can produce a sequence
;; and feed it into a map, for example.</p>
;;
;; <p>For this problem, your goal is to "flatten" a map of hashmaps.
;; Each key in your output map should be the "path"<sup>1</sup> that
;; you would have to take in the original map to get to a value, so
;; for example <code>{1 {2 3}}</code> should result in <code>{[1 2] 3}</code>.
;;
;; You only need to flatten one level of maps: if one of the values is a map,
;; just leave it alone.</p>
;;
;; <p><sup>1</sup> That is, <code>(get-in original [k1 k2])</code> should
;; be the same as <code>(get result [k1 k2])</code></p>.
(defn tree->table
"Flattens a map."
[m]
(reduce (fn [acc [k v]]
(reduce (fn step [p [ke ve]]
(assoc p [k ke] ve))
acc (seq v)))
{} m))
(and (= (tree->table '{a {p 1, q 2}
b {m 3, n 4}})
'{[a p] 1, [a q] 2
[b m] 3, [b n] 4})
(= (tree->table '{[1] {a b c d}
[2] {q r s t u v w x}})
'{[[1] a] b, [[1] c] d,
[[2] q] r, [[2] s] t,
[[2] u] v, [[2] w] x})
(= (tree->table '{m {1 [a b c] 3 nil}})
'{[m 1] [a b c], [m 3] nil}))Pascal’s Trapezoid
;; <p>Write a function that, for any given input vector of numbers,
;; returns an infinite lazy sequence of vectors, where each next one
;; is constructed from the previous following the rules used in
;; <a href="http://en.wikipedia.org/wiki/Pascal's_triangle">Pascal's Triangle</a>.
;; For example, for [3 1 2], the next row is [3 4 3 2].</p>
;;
;; <p>Beware of arithmetic overflow! In clojure (since version 1.3 in 2011),
;; if you use an arithmetic operator like + and the result is too large to
;; fit into a 64-bit integer, an exception is thrown.
;; You can use +' to indicate that you would rather overflow into Clojure's slower,
;; arbitrary-precision bigint.</p>
(defn pascal-triangle [xs]
(letfn [(stepper [xs]
(-> (into [(first xs)] (map #(reduce +' %1N) (partition 2 1 xs)))
(conj (last xs))))]
(iterate stepper xs)))
(and
(= (second (pascal-triangle [2 3 2])) [2 5 5 2])
(= (take 5 (pascal-triangle [1])) [[1] [1 1] [1 2 1] [1 3 3 1] [1 4 6 4 1]])
(= (take 2 (pascal-triangle [3 1 2])) [[3 1 2] [3 4 3 2]])
(= (take 100 (pascal-triangle [2 4 2])) (rest (take 101 (pascal-triangle [2 2])))))The Big Divide
;; <p>Write a function which calculates the sum of all natural numbers
;; under <i>n</i> (first argument) which are evenly divisible by at
;; least one of <i>a</i> and <i>b</i> (second and third argument).
;; Numbers <i>a</i> and <i>b</i> are guaranteed to be
;; <a href="http://en.wikipedia.org/wiki/Coprime">coprimes</a>.</p>
;; <p>Note: Some test cases have a very large <i>n</i>,
;; so the most obvious solution will exceed the time limit.</p>
(defn big-divide [n a b]
(letfn [(n-sum [x limit]
(let [n-val (quot limit x)]
(if (pos? n-val)
(-> (inc n-val) (*' n-val) (/ 2) (*' x) bigint)
0)))]
(let [limit (dec n)]
(-> (+' (n-sum a limit) (n-sum b limit))
(-' (n-sum (*' a b) limit))))))
(and
(= 0 (big-divide 3 17 11))
(= 23 (big-divide 10 3 5))
(= 233168 (big-divide 1000 3 5))
(= "2333333316666668" (str (big-divide 100000000 3 5)))
(= "110389610389889610389610"
(str (big-divide (* 10000 10000 10000) 7 11)))
(= "1277732511922987429116"
(str (big-divide (* 10000 10000 10000) 757 809)))
(= "4530161696788274281"
(str (big-divide (* 10000 10000 1000) 1597 3571))))THERE IS NO PROBLEM 149
Palindromic number
;; A palindromic number is a number that is the same
;; when written forwards or backwards (e.g., 3, 99, 14341).
;; Write a function which takes an integer n, as its only argument,
;; and returns an increasing lazy sequence of all palindromic numbers
;; that are not less than n.
;; The most simple solution will exceed the time limit!
(defn palindromic-num-sieve [n]
(letfn [(num->digits [n]
(loop [x n r []]
(if (< x 10) (into [x] r)
(recur (quot x 10) (into [(mod x 10)] r)))))
(digits->num [xs]
(let [ln (count xs)
last-idx (dec ln)]
(reduce (fn [r idx]
(+' r (bigint (*' (Math/pow 10N idx) (xs (-' last-idx idx))))))
0N (range ln))))
(get-mid-idx [digits]
(bigint (Math/ceil (/ (count digits) 2))))
(mirror-digits [left-digits mid-idx to-replicate]
(if (zero? mid-idx)
left-digits
(into left-digits (reverse (take to-replicate left-digits)))))
(inc-digits [digits pos]
(if (neg? pos)
;; we have to grow the left side
(into [1] (repeat (count digits) 0))
(let [num-at-pos (digits pos)]
(if (< num-at-pos 9)
(assoc digits pos (inc num-at-pos))
(recur (assoc digits pos 0) (dec pos) )))))
(maybe-match-first-last-digits [digits left-side-grown?]
(if left-side-grown?
(assoc digits (dec (count digits)) (first digits))
digits))
(cur-palindrome [pal-number inc-left-side?]
(let [pal (num->digits pal-number)
mid-idx (get-mid-idx pal)
left-digits (if (neg? mid-idx) pal (subvec pal 0 mid-idx))
next-left-digits (if inc-left-side?
(inc-digits left-digits (dec (count left-digits)))
left-digits)
to-replicate (bigint (Math/floor (/ (count pal) 2)))
left-grown? (> (count next-left-digits) (count left-digits))]
(-> (mirror-digits next-left-digits mid-idx to-replicate)
(maybe-match-first-last-digits left-grown?)
digits->num)))
(next-palindrome [pal-number] (cur-palindrome pal-number true))
(ensure-palindrome [cur-num target]
(if (< cur-num 9)
(bigint cur-num)
(let [next-num (cur-palindrome cur-num false)]
(if (< next-num target)
(cur-palindrome cur-num true)
next-num))))]
(iterate next-palindrome (ensure-palindrome n n))))
(and
(= (take 26 (palindromic-num-sieve 0))
[0 1 2 3 4 5 6 7 8 9
11 22 33 44 55 66 77 88 99
101 111 121 131 141 151 161])
(= (take 16 (palindromic-num-sieve 162))
[171 181 191 202
212 222 232 242
252 262 272 282
292 303 313 323])
;;
(= (take 6 (palindromic-num-sieve 1234550000))
[1234554321 1234664321 1234774321
1234884321 1234994321 1235005321])
(= (first (palindromic-num-sieve (* 111111111 111111111)))
(* 111111111 111111111))
(= (set (take 199 (palindromic-num-sieve 0)))
(set (map #(first (palindromic-num-sieve %)) (range 0 10000))))
(= true
(apply < (take 6666 (palindromic-num-sieve 9999999))))
(= (nth (palindromic-num-sieve 0) 10101)
9102019))Language of a DFA TBD: recursion -> iteration to pass test
;; http://en.wikipedia.org/wiki/Deterministic_finite_automaton
;; http://en.wikipedia.org/wiki/Regular_language
;; A deterministic finite automaton (DFA) is an abstract machine that recognizes
;; a regular language.
;; Usually a DFA is defined by a 5-tuple, but instead we'll use a map with 5 keys:
;; -:states</var> is the set of states for the DFA.
;; -:alphabet</var> is the set of symbols included in the language recognized by the DFA.
;; -:start</var> is the start state of the DFA.
;; -:accepts</var> is the set of accept states in the DFA.
;; -:transitions</var> is the transition function for the DFA, mapping
;; :states :alphabet onto :states.
;; Write a function that takes as input a DFA definition (as described above)
;; and returns a sequence enumerating all strings in the language recognized by the DFA.
;; Note: Although the DFA itself is finite and only recognizes finite-length strings
;; it can still recognize an infinite set of finite-length strings.
;; And because stack space is finite, make sure you don't get stuck in an infinite loop
;; that's not producing results every so often!
(defn analyze-dfa [g]
(letfn [(accept? [u]
(contains? (:accepts g) u))
(visit [[w u] transitions p r]
(lazy-seq
(let [next-states (seq (get-in g [:transitions u]))
accepted? (accept? u)
cur-path (if (nil? w)
p
(conj p w))
ret (if accepted?
(conj r cur-path)
r)]
(if (empty? next-states)
(map (fn [coll] (apply str coll)) ret)
(mapcat (fn [[sym v]]
(visit [sym v] (dissoc transitions v) cur-path ret))
next-states)))))]
(visit [nil (:start g)] (:transitions g) [] [])))
(and
(= #{"a" "ab" "abc"}
(set (analyze-dfa '{:states #{q0 q1 q2 q3}
:alphabet #{a b c}
:start q0
:accepts #{q1 q2 q3}
:transitions {q0 {a q1}
q1 {b q2}
q2 {c q3}}})))
(= #{"hi" "hey" "hello"}
(set (analyze-dfa '{:states #{q0 q1 q2 q3 q4 q5 q6 q7}
:alphabet #{e h i l o y}
:start q0
:accepts #{q2 q4 q7}
:transitions {q0 {h q1}
q1 {i q2, e q3}
q3 {l q5, y q4}
q5 {l q6}
q6 {o q7}}})))
(= (set (let [ss "vwxyz"] (for [i ss, j ss, k ss, l ss] (str i j k l))))
(set (analyze-dfa '{:states #{q0 q1 q2 q3 q4}
:alphabet #{v w x y z}
:start q0
:accepts #{q4}
:transitions {q0 {v q1, w q1, x q1, y q1, z q1}
q1 {v q2, w q2, x q2, y q2, z q2}
q2 {v q3, w q3, x q3, y q3, z q3}
q3 {v q4, w q4, x q4, y q4, z q4}}})))
(let [res (take 2000 (analyze-dfa '{:states #{q0 q1}
:alphabet #{0 1}
:start q0
:accepts #{q0}
:transitions {q0 {0 q0, 1 q1}
q1 {0 q1, 1 q0}}}))]
(and (every? (partial re-matches #"0*(?:10*10*)*") res)
(= res (distinct res))))
(let [res (take 2000 (analyze-dfa '{:states #{q0 q1}
:alphabet #{n m}
:start q0
:accepts #{q1}
:transitions {q0 {n q0, m q1}}}))]
(and (every? (partial re-matches #"n*m") res)
(= res (distinct res))))
(let [res (take 2000 (analyze-dfa '{:states #{q0 q1 q2 q3 q4 q5 q6 q7 q8 q9}
:alphabet #{i l o m p t}
:start q0
:accepts #{q5 q8}
:transitions {q0 {l q1}
q1 {i q2, o q6}
q2 {m q3}
q3 {i q4}
q4 {t q5}
q6 {o q7}
q7 {p q8}
q8 {l q9}
q9 {o q6}}}))]
(and (every? (partial re-matches #"limit|(?:loop)+") res)
(= res (distinct res)))))