Project Euler
Some random answers for some Project Euler questions that I've done when I was bored.
(ns euler.problems
(:require [clojure.core.match :refer [match]]))
(defn problem-1
"If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000."
[]
;; Faster but not as pretty :P
;; (loop [curr 999
;; sum 0]
;; (if (zero? curr)
;; sum
;; (let [divisible-by-3-5? (or (zero? (rem curr 3))
;; (zero? (rem curr 5)))]
;; (recur (dec curr) (if divisible-by-3-5?
;; (+ sum curr)
;; sum)))))
(->> (range 1000)
(filter #(some zero? (map (partial rem %) [3 5])))
(reduce +)))
(defn problem-2
"Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms."
[]
(letfn [(fib
([n] (fib n 0 1))
([n x y] (match [n]
[0] x
[1] y
[_] (recur (dec n) y (+ x y)))))]
(->> (for [n (range)
:let [fib-n (fib n)]
:while (> 4000000 fib-n)
:when even?]
fib-n)
(reduce +))))
(defn problem-3
"The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?"
[]
(letfn
[(multiple? [n div]
(zero? (mod n div)))
(prime-factors
([n]
(prime-factors n 2 (transient [])))
([n candidate acc]
(cond (<= n 1) (reverse (persistent! acc))
(multiple? n candidate) (recur (/ n candidate) candidate (conj! acc candidate))
:else (recur n (inc candidate) acc))))]
(apply max (prime-factors 600851475143))))
(defn problem-4
"A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.
Find the largest palindrome made from the product of two 3-digit numbers."
[]
(letfn
[(palindrome? ([num] (= (str num)
(apply str (reverse (str num))))))]
(->> (for [x (range 1000)
y (range 1000)
:let [product (* x y)]
:when (palindrome? product)]
product)
(apply max))))
(defn problem-5
"2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?"
[]
(let [divisors (range 19 10 -1)]
(->> (for [n (iterate (partial + 20) 20)
:when (do (when (zero? (rem n 10000000)) (println n))
(every? zero? (map (fn [divisor] (rem n divisor)) divisors)))]
n)
(first))))
(defn problem-6
"The sum of the squares of the first ten natural numbers is,
12 + 22 + .j.. + 102 = 385
The square of the sum of the first ten natural numbers is,
(1 + 2 + ... + 10)2 = 552 = 3025
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 − 385 = 2640.
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum."
[]
(letfn [(sum-of-squares
([up-to] (->> (range 1 (inc up-to))
(map (fn [n] (* n n)))
(reduce +))))
(square-of-sum
([up-to] (Math/pow
(->> (range 1 (inc up-to))
(reduce +))
2)))]
(- (square-of-sum 100) (sum-of-squares 100))))
(defn problem-7
"By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
What is the 10 001st prime number?"
[]
(letfn [(isPrime?
([n] (isPrime? n 2))
([n start] (cond
(> start (Math/sqrt n)) (> n 1)
(< (mod n start) 1) false
:else (recur n (inc start)))))]
(->> (range)
(filter isPrime?)
(take 10001)
(last))))
(defn problem-8
"The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832. 73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450
Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?"
[]
(let [big-number-string "7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450"
number-chars (clojure.string/split big-number-string #"")]
(->> number-chars
(partition-all 13 1)
(pmap (fn [list]
(reduce * (map read-string list))))
(apply max))))
(defn problem-9
"A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
a2 + b2 = c2
For example, 32 + 42 = 9 + 16 = 25 = 52.
There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc."
[]
(->> (let [triplets
(for [a (range 1 1000)
b (range (inc a) 1000)
:let [c-squared (+ (Math/pow b 2) (Math/pow a 2))
c-float (Math/sqrt c-squared)
triplet [a b (int c-float)]
a-squared (Math/pow a 2)
b-squared (Math/pow b 2)]
:when (and
(= (float (int c-float)) c-float)
(> c-float b)
(> b a)
(= (+ a-squared b-squared) c-squared))]
(do
#_(println triplet)
triplet))]
(->> triplets
(filter (fn [triple] (= 1000 (reduce + triple))))
(first)
(reduce *)))))
(defn problem-10
"The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million."
[]
(letfn [(prime?
([n] (prime? n 2))
([n start] (cond
(> start (Math/sqrt n)) (> n 1)
(< (mod n start) 1) false
:else (recur n (inc start)))))]
(->> (map inc (range 2000000))
(pmap #(if (prime? %) % 0))
(reduce +))))
(defn problem-11
"In the 20×20 grid below, four numbers along a diagonal line have been marked in red.
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91
22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80
24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50
32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70
67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21
24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72
21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95
78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48
The product of these numbers is 26 × 63 × 78 × 14 = 1788696.
What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20×20 grid?"
[]
(let [matrix-string "08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08\n 49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00\n 81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65\n 52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91\n 22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80\n 24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50\n 32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70\n 67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21\n 24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72\n 21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95\n 78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92\n 16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57\n 86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58\n 19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40\n 04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66\n 88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69\n 04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36\n 20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16\n 20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54\n 01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48"
number-list (->> (clojure.string/split-lines matrix-string)
(map (fn [row] (clojure.string/trim row)))
(map (fn [row] (->> (clojure.string/split row #" ")
(map (fn [string] (int string)))
(filter number?))))
(flatten))
number-matrix (partition 20 number-list)
rows (for [y (range 0 20)
:let [row (nth number-matrix y)]]
row)
columns (for [x (range 0 20)
:let [column (take-nth 20 (drop x number-list))]]
column)
diagonal-lr-4s (->> (for [x (range 0 (reduce inc number-list))
:let [diagonal (take-nth 21 (drop x number-list))]
:when (<= (mod x 20) 16)]
(partition 4 1 diagonal))
(apply concat))
diagonal-rl-4s (->> (for [x (range 0 (reduce inc number-list))
:let [diagonal (take-nth 19 (drop x number-list))]
:when (>= (mod x 20) 3)]
(partition 4 1 diagonal))
(apply concat))
horizontal-4s (->> (for [row rows
:let [fours (partition 4 1 row)]]
fours)
(apply concat))
vertical-4s (->> (for [column columns
:let [fours (partition 4 1 column)]]
fours)
(apply concat))
all-4s (->> (concat diagonal-lr-4s diagonal-rl-4s horizontal-4s vertical-4s))
products (->> all-4s
(map (fn [four] (reduce * four))))]
(apply max products)))
(defn problem-12
"the sequence of triangle numbers is generated by adding the natural numbers. so the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. the first ten terms would be:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
let us list the factors of the first seven triangle numbers:
1: 1
3: 1,3
6: 1,2,3,6
10: 1,2,5,10
15: 1,3,5,15
21: 1,3,7,21
28: 1,2,4,7,14,28
we can see that 28 is the first triangle number to have over five divisors.
what is the value of the first triangle number to have over five hundred divisors?"
[]
(let [triangle-numbers (->> (range)
(pmap #(reduce + (range 0 %))))
factors (fn [n] (into (sorted-set)
(reduce concat
(for [x (range 1 (inc (Math/sqrt n)))
:when (zero? (rem n x))]
[x (/ n x)]))))]
(->> triangle-numbers
(pmap (fn [tri] {:triangle-number tri :factors (factors tri)}))
(filter #(> (count (:factors %)) 500))
(first)
(:triangle-number))))
(defn problem-13
[]
(let [number-string "37107287533902102798797998220837590246510135740250\n46376937677490009712648124896970078050417018260538\n74324986199524741059474233309513058123726617309629\n91942213363574161572522430563301811072406154908250\n23067588207539346171171980310421047513778063246676\n89261670696623633820136378418383684178734361726757\n28112879812849979408065481931592621691275889832738\n44274228917432520321923589422876796487670272189318\n47451445736001306439091167216856844588711603153276\n70386486105843025439939619828917593665686757934951\n62176457141856560629502157223196586755079324193331\n64906352462741904929101432445813822663347944758178\n92575867718337217661963751590579239728245598838407\n58203565325359399008402633568948830189458628227828\n80181199384826282014278194139940567587151170094390\n35398664372827112653829987240784473053190104293586\n86515506006295864861532075273371959191420517255829\n71693888707715466499115593487603532921714970056938\n54370070576826684624621495650076471787294438377604\n53282654108756828443191190634694037855217779295145\n36123272525000296071075082563815656710885258350721\n45876576172410976447339110607218265236877223636045\n17423706905851860660448207621209813287860733969412\n81142660418086830619328460811191061556940512689692\n51934325451728388641918047049293215058642563049483\n62467221648435076201727918039944693004732956340691\n15732444386908125794514089057706229429197107928209\n55037687525678773091862540744969844508330393682126\n18336384825330154686196124348767681297534375946515\n80386287592878490201521685554828717201219257766954\n78182833757993103614740356856449095527097864797581\n16726320100436897842553539920931837441497806860984\n48403098129077791799088218795327364475675590848030\n87086987551392711854517078544161852424320693150332\n59959406895756536782107074926966537676326235447210\n69793950679652694742597709739166693763042633987085\n41052684708299085211399427365734116182760315001271\n65378607361501080857009149939512557028198746004375\n35829035317434717326932123578154982629742552737307\n94953759765105305946966067683156574377167401875275\n88902802571733229619176668713819931811048770190271\n25267680276078003013678680992525463401061632866526\n36270218540497705585629946580636237993140746255962\n24074486908231174977792365466257246923322810917141\n91430288197103288597806669760892938638285025333403\n34413065578016127815921815005561868836468420090470\n23053081172816430487623791969842487255036638784583\n11487696932154902810424020138335124462181441773470\n63783299490636259666498587618221225225512486764533\n67720186971698544312419572409913959008952310058822\n95548255300263520781532296796249481641953868218774\n76085327132285723110424803456124867697064507995236\n37774242535411291684276865538926205024910326572967\n23701913275725675285653248258265463092207058596522\n29798860272258331913126375147341994889534765745501\n18495701454879288984856827726077713721403798879715\n38298203783031473527721580348144513491373226651381\n34829543829199918180278916522431027392251122869539\n40957953066405232632538044100059654939159879593635\n29746152185502371307642255121183693803580388584903\n41698116222072977186158236678424689157993532961922\n62467957194401269043877107275048102390895523597457\n23189706772547915061505504953922979530901129967519\n86188088225875314529584099251203829009407770775672\n11306739708304724483816533873502340845647058077308\n82959174767140363198008187129011875491310547126581\n97623331044818386269515456334926366572897563400500\n42846280183517070527831839425882145521227251250327\n55121603546981200581762165212827652751691296897789\n32238195734329339946437501907836945765883352399886\n75506164965184775180738168837861091527357929701337\n62177842752192623401942399639168044983993173312731\n32924185707147349566916674687634660915035914677504\n99518671430235219628894890102423325116913619626622\n73267460800591547471830798392868535206946944540724\n76841822524674417161514036427982273348055556214818\n97142617910342598647204516893989422179826088076852\n87783646182799346313767754307809363333018982642090\n10848802521674670883215120185883543223812876952786\n71329612474782464538636993009049310363619763878039\n62184073572399794223406235393808339651327408011116\n66627891981488087797941876876144230030984490851411\n60661826293682836764744779239180335110989069790714\n85786944089552990653640447425576083659976645795096\n66024396409905389607120198219976047599490197230297\n64913982680032973156037120041377903785566085089252\n16730939319872750275468906903707539413042652315011\n94809377245048795150954100921645863754710598436791\n78639167021187492431995700641917969777599028300699\n15368713711936614952811305876380278410754449733078\n40789923115535562561142322423255033685442488917353\n44889911501440648020369068063960672322193204149535\n41503128880339536053299340368006977710650566631954\n81234880673210146739058568557934581403627822703280\n82616570773948327592232845941706525094512325230608\n22918802058777319719839450180888072429661980811197\n77158542502016545090413245809786882778948721859617\n72107838435069186155435662884062257473692284509516\n20849603980134001723930671666823555245252804609722\n53503534226472524250874054075591789781264330331690"
numbers (->> (clojure.string/split-lines number-string)
(map clojure.string/trim)
(map read-string))]
(->> numbers
(reduce +)
(str)
(take 10)
(clojure.string/join ""))))
(defn problem-14
"The following iterative sequence is defined for the set of positive integers:
n → n/2 (n is even)
n → 3n + 1 (n is odd)
Using the rule above and starting with 13, we generate the following sequence:
13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.
Which starting number, under one million, produces the longest chain?
NOTE: Once the chain starts the terms are allowed to go above one million."
[]
(letfn [(collatz
([n]
(collatz n (transient [])))
([n t]
(cond (= 1 n) (do (conj! t 1)
(persistent! t))
(even? n) (recur (/ n 2) (conj! t n))
(odd? n) (recur (inc (* 3 n)) (conj! t n)))))]
(->> (range 1 1000001)
(pmap (fn [n] (do
(when (zero? (rem n 10000))
(println n))
{:key n :count (count (collatz n))})))
(apply max-key :count)
(:key))))
(defn problem-16
"2^15 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
What is the sum of the digits of the number 2^1000?"
[]
(->> (repeat 2)
(take 1000)
(reduce *')
(str)
(map (comp read-string str))
(reduce +)))
(defn problem-17
"If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.
If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words, how many letters would be used?
NOTE: Do not count spaces or hyphens. For example, 342 (three hundred and forty-two) contains 23 letters and 115 (one hundred and fifteen) contains 20 letters. The use of \"and\" when writing out numbers is in compliance with British usage."
[]
(let [number-to-english-map {0 ""
1 "one"
2 "two"
3 "three"
4 "four"
5 "five"
6 "six"
7 "seven"
8 "eight"
9 "nine"
10 "ten"
11 "eleven"
12 "twelve"
13 "thirteen"
14 "fourteen"
15 "fifteen"
16 "sixteen"
17 "seventeen"
18 "eighteen"
19 "nineteen"
20 "twenty"
30 "thirty"
40 "forty"
50 "fifty"
60 "sixty"
70 "seventy"
80 "eighty"
90 "ninety"
100 "hundred"
1000 "thousand"}]
(letfn [(convert-vector-to-english
([vec]
(convert-vector-to-english vec ""))
([vec carry]
(let [next (into [] (drop 1 vec))]
(match [vec]
[[ones]]
(str carry
(number-to-english-map ones))
[[tens ones]]
(str carry
(let [sum (+ (* tens 10) ones)]
(if (< sum 20)
(number-to-english-map sum)
(str (number-to-english-map (* tens 10))
(when (not (zero? ones))
(str " " (number-to-english-map ones)))))))
[[hundreds tens ones]]
(recur next
(str carry
(when (pos-int? hundreds)
(str (number-to-english-map hundreds) " hundred"))
(when (some pos-int? [tens ones])
" and ")))
[[thousands hundreds tens ones]]
(recur next
(str (when (pos-int? thousands)
(str (number-to-english-map thousands) " thousand"))
(when (some pos-int? [hundreds tens ones])
" ")))))))
(number-to-english
([n] (let [number-vec (into [] (map (comp read-string str) (str n)))]
(convert-vector-to-english number-vec))))]
(->> (range 1 1001)
(map number-to-english)
(clojure.string/join " ")
(map str)
(filter #(not= " " %))
(count)))))