Next chunk of solutions

p011.groovy

@@ -0,0 +1,46 @@
+table = """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"""
+
+int[][] matrix = new int[26][26]
+(0..25).each { i -> (0..25).each { j -> matrix[i][j] = 1}}
+
+table.split('\n').eachWithIndex { line, i ->
+    row = line.split(' ')*.toInteger()
+    row.eachWithIndex { item, j ->
+        matrix[i+3][j+3] = item
+    }
+}
+
+max = 0
+
+(0..19).each { i ->
+    (0..19).each { j ->
+        prod = matrix[i+3][j+3] * matrix[i+3][j+4] * matrix[i+3][j+5] * matrix[i+3][j+6]
+        if (max < prod) max = prod
+        prod = matrix[i+3][j+3] * matrix[i+4][j+3] * matrix[i+5][j+3] * matrix[i+6][j+3]
+        if (max < prod) max = prod
+        prod = matrix[i+3][j+3] * matrix[i+4][j+4] * matrix[i+5][j+5] * matrix[i+6][j+6]
+        if (max < prod) max = prod
+        prod = matrix[i+3][j+3] * matrix[i+4][j+2] * matrix[i+5][j+1] * matrix[i+6][j+0]
+        if (max < prod) max = prod
+    }
+}
+println max

p012.erl

@@ -0,0 +1,67 @@
+%% Problem
+%% ---------------------
+%% 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?
+%% ---------------------
+
+-module(p012).
+-export([solve1/0, solve2/0]).
+-include_lib("eunit/include/eunit.hrl").
+
+%%
+%% Brute force solution
+%%
+solve2() -> triangular(7, 0).
+
+triangular(N, Result) when Result > 500 -> N*(N-1) div 2;
+triangular(N, _) -> triangular(N+1, factors(N*(N+1) div 2)).
+
+%%
+%% Returns number of divisors of integer N
+%%
+factors(N) -> factors(N, trunc(math:sqrt(N))).
+
+factors(N, L) when L*L == N -> factors(N, L+1) - 1;
+factors(N, L) -> 2 * length([ X || X <- lists:seq(1,L), N rem X =:= 0 ]).
+
+%%
+%% Faster solution uses the fact that
+%% D(t) = D(n/2)D(n+1) (n is even) or D(t) = D(n)*D((n+1)/2) (n is odd)
+%% Also reusing D(n) from previous iteration
+%%
+solve1() -> tri(2, 1, 1).
+
+tri(N, _, Result) when Result > 500 -> N*(N-1) div 2;
+tri(N, D, _) ->
+    M = case N rem 2 == 0 of
+        true -> N+1;
+        false -> (N+1) div 2
+    end,
+    D1 = factors(M),
+    tri(N+1, D1, D*D1).
+
+
+% Tests
+
+factors_25_test() ->
+    ?assertEqual(3, factors(25)).
+
+factors_28_test() ->
+    ?assertEqual(6, factors(28)).

p013.erl

@@ -0,0 +1,114 @@
+%% Problem
+%% ---------------------
+%% Find the first ten digits of the sum of one-hundred 50-digit numbers.
+%% ---------------------
+
+-module(p013).
+-export([solve/0]).
+
+numbers() -> [
+37107287533902102798797998220837590246510135740250,
+46376937677490009712648124896970078050417018260538,
+74324986199524741059474233309513058123726617309629,
+91942213363574161572522430563301811072406154908250,
+23067588207539346171171980310421047513778063246676,
+89261670696623633820136378418383684178734361726757,
+28112879812849979408065481931592621691275889832738,
+44274228917432520321923589422876796487670272189318,
+47451445736001306439091167216856844588711603153276,
+70386486105843025439939619828917593665686757934951,
+62176457141856560629502157223196586755079324193331,
+64906352462741904929101432445813822663347944758178,
+92575867718337217661963751590579239728245598838407,
+58203565325359399008402633568948830189458628227828,
+80181199384826282014278194139940567587151170094390,
+35398664372827112653829987240784473053190104293586,
+86515506006295864861532075273371959191420517255829,
+71693888707715466499115593487603532921714970056938,
+54370070576826684624621495650076471787294438377604,
+53282654108756828443191190634694037855217779295145,
+36123272525000296071075082563815656710885258350721,
+45876576172410976447339110607218265236877223636045,
+17423706905851860660448207621209813287860733969412,
+81142660418086830619328460811191061556940512689692,
+51934325451728388641918047049293215058642563049483,
+62467221648435076201727918039944693004732956340691,
+15732444386908125794514089057706229429197107928209,
+55037687525678773091862540744969844508330393682126,
+18336384825330154686196124348767681297534375946515,
+80386287592878490201521685554828717201219257766954,
+78182833757993103614740356856449095527097864797581,
+16726320100436897842553539920931837441497806860984,
+48403098129077791799088218795327364475675590848030,
+87086987551392711854517078544161852424320693150332,
+59959406895756536782107074926966537676326235447210,
+69793950679652694742597709739166693763042633987085,
+41052684708299085211399427365734116182760315001271,
+65378607361501080857009149939512557028198746004375,
+35829035317434717326932123578154982629742552737307,
+94953759765105305946966067683156574377167401875275,
+88902802571733229619176668713819931811048770190271,
+25267680276078003013678680992525463401061632866526,
+36270218540497705585629946580636237993140746255962,
+24074486908231174977792365466257246923322810917141,
+91430288197103288597806669760892938638285025333403,
+34413065578016127815921815005561868836468420090470,
+23053081172816430487623791969842487255036638784583,
+11487696932154902810424020138335124462181441773470,
+63783299490636259666498587618221225225512486764533,
+67720186971698544312419572409913959008952310058822,
+95548255300263520781532296796249481641953868218774,
+76085327132285723110424803456124867697064507995236,
+37774242535411291684276865538926205024910326572967,
+23701913275725675285653248258265463092207058596522,
+29798860272258331913126375147341994889534765745501,
+18495701454879288984856827726077713721403798879715,
+38298203783031473527721580348144513491373226651381,
+34829543829199918180278916522431027392251122869539,
+40957953066405232632538044100059654939159879593635,
+29746152185502371307642255121183693803580388584903,
+41698116222072977186158236678424689157993532961922,
+62467957194401269043877107275048102390895523597457,
+23189706772547915061505504953922979530901129967519,
+86188088225875314529584099251203829009407770775672,
+11306739708304724483816533873502340845647058077308,
+82959174767140363198008187129011875491310547126581,
+97623331044818386269515456334926366572897563400500,
+42846280183517070527831839425882145521227251250327,
+55121603546981200581762165212827652751691296897789,
+32238195734329339946437501907836945765883352399886,
+75506164965184775180738168837861091527357929701337,
+62177842752192623401942399639168044983993173312731,
+32924185707147349566916674687634660915035914677504,
+99518671430235219628894890102423325116913619626622,
+73267460800591547471830798392868535206946944540724,
+76841822524674417161514036427982273348055556214818,
+97142617910342598647204516893989422179826088076852,
+87783646182799346313767754307809363333018982642090,
+10848802521674670883215120185883543223812876952786,
+71329612474782464538636993009049310363619763878039,
+62184073572399794223406235393808339651327408011116,
+66627891981488087797941876876144230030984490851411,
+60661826293682836764744779239180335110989069790714,
+85786944089552990653640447425576083659976645795096,
+66024396409905389607120198219976047599490197230297,
+64913982680032973156037120041377903785566085089252,
+16730939319872750275468906903707539413042652315011,
+94809377245048795150954100921645863754710598436791,
+78639167021187492431995700641917969777599028300699,
+15368713711936614952811305876380278410754449733078,
+40789923115535562561142322423255033685442488917353,
+44889911501440648020369068063960672322193204149535,
+41503128880339536053299340368006977710650566631954,
+81234880673210146739058568557934581403627822703280,
+82616570773948327592232845941706525094512325230608,
+22918802058777319719839450180888072429661980811197,
+77158542502016545090413245809786882778948721859617,
+72107838435069186155435662884062257473692284509516,
+20849603980134001723930671666823555245252804609722,
+53503534226472524250874054075591789781264330331690].
+
+%%
+%% Dumb solution
+%%
+solve() -> lists:sum(numbers()).

p014.erl

@@ -0,0 +1,34 @@
+%% Problem
+%% ---------------------
+%% 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?
+%% ---------------------
+
+-module(p014).
+-export([solve/0]).
+-include_lib("eunit/include/eunit.hrl").
+
+%%
+%% Brute force solution (still relatively fast)
+%%
+solve() -> lists:max([ {cl(N), N} || N <- lists:seq(2,1000000) ]).
+
+%%
+%% Chain length
+%%
+cl(1) -> 1; 
+cl(N) when N rem 2 =:= 0 -> 1 + cl(N div 2); 
+cl(N) -> 1 + cl(3 * N + 1).
+
+cl_test() -> ?assertEqual(10, cl(13)).

p015.erl

@@ -0,0 +1,12 @@
+%% Problem
+%% ---------------------
+%% Starting in the top left corner of a 2x2 grid, there are 6 routes
+%% (without backtracking) to the bottom right corner.
+%%
+%% How many routes are there through a 20x20 grid?
+%% ---------------------
+
+-module(p015).
+-export([solve/0]).
+
+solve() -> mymath:c(40, 20).

p016.erl

@@ -0,0 +1,11 @@
+%% Problem
+%% ---------------------
+%% 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?
+%% ---------------------
+
+-module(p016).
+-export([solve/0]).
+
+solve() -> mymath:ds(trunc(math:pow(2,1000))).

p018.groovy

@@ -0,0 +1,20 @@
+// inspired by
+// http://blog.dreamshire.com/2009/04/01/project-euler-problem-18-solution
+
+// to solve problem 67 replace file name to triangle67.txt
+
+def max(x, y) {
+    x < y ? y : x
+}
+
+triangle = []
+new File('triangle18.txt').eachLine { line ->
+    triangle << line.split(/ /)*.toInteger()
+}
+(triangle.size()-1).downto(1) { i ->
+    0.upto(i-1) { j ->
+        triangle[i-1][j] += max(triangle[i][j], triangle[i][j+1])
+    }
+
+}
+println triangle[0][0]

p020.erl

@@ -0,0 +1,9 @@
+%% Problem
+%% ---------------------
+%% Find the sum of digits in 100!
+%% ---------------------
+
+-module(p020).
+-export([solve/0]).
+
+solve() -> mymath:ds(mymath:factorial(100)).

p021.erl

@@ -0,0 +1,31 @@
+%% Problem
+%% ---------------------
+%% Evaluate the sum of all the amicable numbers under 10000.
+%% ---------------------
+
+-module(p021).
+-export([solve/0]).
+
+
+%% Brute force works relatively fast.
+%%
+solve() ->
+    Tuples = candidates(),
+    Amis = [ {N, D} || {N, D} <- Tuples, {N1, D1} <- Tuples, N1 =:= D, D1 =:= N ],
+    lists:foldl(fun({N, _}, Sum) -> N + Sum end, 0, Amis).
+
+candidates() ->
+    lists:filter(fun({N, D}) -> (D > 1) and (N =/= D) end, all_tuples()).
+
+all_tuples() ->
+    [ {N, d(N)} || N <- lists:seq(2, 10000-1) ].
+
+d(N) ->
+    lists:sum(prop_devisors(N)).
+
+prop_devisors(N) ->
+    [ M || M <- lists:seq(1, N-1), N rem M =:= 0 ].
+
+
+% See also sigma function
+% http://mathschallenge.net/index.php?section=faq&ref=number/sum_of_divisors

p022.groovy

@@ -0,0 +1,13 @@
+def wordValue(word) {
+    word.bytes.inject(0) { sum, i -> sum + i - 64 }
+}
+
+names = []
+new File('names.txt').eachLine { line ->
+    line.split(/,/).each { names << it[1..-2] }
+}
+
+result = 0
+names.sort().eachWithIndex { name, i -> result += (i + 1) * wordValue(name) }
+
+println result