Euler problems
These are the solutions to the problems listed on Project Euler
It is recommended you try them yourself before looking at the solutions as these form good exercises for improving your Haskell-hu.
Problem 1
Add all the natural numbers below 1000 that are multiples of 3 or 5.
Solution:
problem_1 = sum [ x | x <- [1..1000], x `mod` 3 == 0, x `mod` 5 == 0]
Problem 2
Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million.
Solution:
problem_2 = sum [ x | x <- fibs, x `mod` 2 == 0]
where fibs = 1 : 1 : zipWith (+) fibs (tail fibs)
Problem 3
Find the largest prime factor of 317584931803.
Solution:
problem_3 = foldr max 0 [ x | x <- [1..(round $ sqrt c)], c `mod` x == 0]
where c = 317584931803
Problem 4
Find the largest palindrome made from the product of two 3-digit numbers.
Solution:
problem_4 = foldr max 0 [ x | y <- [100..999], z <- [100..999], let x = y * z, let s = show x, s == reverse s]
Problem 5
What is the smallest number divisible by each of the numbers 1 to 20?
Solution:
problem_5 = head [ x | x <- [2520,5040..], all (\y -> x `mod` y == 0) [1..20]]
Problem 6
What is the difference between the sum of the squares and the square of the sums?
Solution:
problem_6 = sum [ x^2 | x <- [1..100]] - (sum [1..100])^2
Problem 7
Find the 10001st prime.
Solution:
problem_7 = head $ drop 10000 primes
where primes = 2:3:..
Problem 8
Discover the largest product of five consecutive digits in the 1000-digit number.
Solution:
problem_8 = undefined
Problem 9
Find the only Pythagorean triplet, {a, b, c}, for which a + b + c = 1000.
Solution:
problem_9 = head [a*b*c | a <- [1..500], b <- [a..500], c <- [1..(1000-a-b)], a + b + c == 1000, a^2 + b^2 == c^2]
Problem 10
Calculate the sum of all the primes below one million.
Solution:
problem_10 = sum [ p | p <- primes, p < 1000000 ]
Problem 11
What is the greatest product of four numbers on the same straight line in the 20 by 20 grid?
Solution:
problem_11 = undefined
Problem 12
What is the first triangle number to have over five-hundred divisors?
Solution:
problem_12 = undefined
Problem 13
Find the first ten digits of the sum of one-hundred 50-digit numbers.
Solution:
problem_13 = undefined
Problem 14
Find the longest sequence using a starting number under one million.
Solution:
problem_14 = undefined
Problem 15
Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner?
Solution:
problem_15 = undefined
Problem 16
What is the sum of the digits of the number 21000?
Solution:
problem_16 = undefined
Problem 17
How many letters would be needed to write all the numbers in words from 1 to 1000?
Solution:
problem_17 = undefined
Problem 18
Find the maximum sum travelling from the top of the triangle to the base.
Solution:
problem_18 = undefined
Problem 19
How many Sundays fell on the first of the month during the twentieth century?
Solution:
problem_19 = undefined
Problem 20
Find the sum of digits in 100!
Solution:
problem_20 = undefined
Problem 21
Evaluate the sum of all amicable pairs under 10000.
Solution:
problem_21 = undefined
Problem 22
What is the total of all the name scores in the file of first names?
Solution:
problem_22 = undefined
Problem 23
Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
Solution:
problem_23 = undefined
Problem 24
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?
Solution:
problem_24 = undefined
Problem 25
What is the first term in the Fibonacci sequence to contain 1000 digits?
Solution:
problem_25 = undefined
Problem 26
Find the value of d < 1000 for which 1/d contains the longest recurring cycle.
Solution:
problem_26 = undefined
Problem 27
Find a quadratic formula that produces the maximum number of primes for consecutive values of n.
Solution:
problem_27 = undefined
Problem 28
What is the sum of both diagonals in a 1001 by 1001 spiral?
Solution:
problem_28 = undefined
Problem 29
How many distinct terms are in the sequence generated by ab for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?
Solution:
problem_29 = undefined
Problem 30
Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.
Solution:
problem_30 = undefined
Problem 31
Investigating combinations of English currency denominations.
Solution:
problem_31 = undefined
Problem 32
Find the sum of all numbers that can be written as pandigital products.
Solution:
problem_32 = undefined
Problem 33
Discover all the fractions with an unorthodox cancelling method.
Solution:
problem_33 = undefined
Problem 34
Find the sum of all numbers which are equal to the sum of the factorial of their digits.
Solution:
problem_34 = undefined
Problem 35
How many circular primes are there below one million?
Solution:
problem_35 = undefined
Problem 36
Find the sum of all numbers less than one million, which are palindromic in base 10 and base 2.
Solution:
problem_36 = undefined
Problem 37
Find the sum of all eleven primes that are both truncatable from left to right and right to left.
Solution:
problem_37 = undefined
Problem 38
What is the largest 1 to 9 pandigital that can be formed by multiplying a fixed number by 1, 2, 3, ... ?
Solution:
problem_38 = undefined
Problem 39
If p is the perimeter of a right angle triangle, {a, b, c}, which value, for p ≤ 1000, has the most solutions?
Solution:
problem_39 = undefined
Problem 40
Finding the nth digit of the fractional part of the irrational number.
Solution:
problem_40 = undefined
Problem 41
What is the largest n-digit pandigital prime that exists?
Solution:
problem_41 = undefined
Problem 42
How many triangle words can you make using the list of common English words?
Solution:
problem_42 = undefined
Problem 43
Find the sum of all pandigital numbers with an unusual sub-string divisibility property.
Solution:
problem_43 = undefined
Problem 44
Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.
Solution:
problem_44 = undefined
Problem 45
After 40755, what is the next triangle number that is also pentagonal and hexagonal?
Solution:
problem_45 = undefined
Problem 46
What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
Solution:
problem_46 = undefined
Problem 47
Find the first four consecutive integers to have four distinct primes factors.
Solution:
problem_47 = undefined
Problem 48
Find the last ten digits of 11 + 22 + ... + 10001000.
Solution:
problem_48 = undefined
Problem 49
Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other.
Solution:
problem_49 = undefined
Problem 50
Which prime, below one-million, can be written as the sum of the most consecutive primes?
Solution:
problem_50 = undefined
Problem 51
Find the smallest prime which, by changing the same part of the number, can form eight different primes.
Solution:
problem_51 = undefined
Problem 52
Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.
Solution:
problem_52 = undefined
Problem 53
How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?
Solution:
problem_53 = undefined
Problem 54
How many hands did player one win in the game of poker?
Solution:
problem_54 = undefined
Problem 55
How many Lychrel numbers are there below ten-thousand?
Solution:
problem_55 = undefined
Problem 56
Considering natural numbers of the form, ab, finding the maximum digital sum.
Solution:
problem_56 = undefined
Problem 57
Investigate the expansion of the continued fraction for the square root of two.
Solution:
problem_57 = undefined
Problem 58
Investigate the number of primes that lie on the diagonals of the spiral grid.
Solution:
problem_58 = undefined
Problem 59
Using a brute force attack, can you decrypt the cipher using XOR encryption?
Solution:
problem_59 = undefined
Problem 60
Find a set of five primes for which any two primes concatenate to produce another prime.
Solution:
problem_60 = undefined
Problem 61
Find the sum of the only set of six 4-digit figurate numbers with a cyclic property.
Solution:
problem_61 = undefined
Problem 62
Find the smallest cube for which exactly five permutations of its digits are cube.
Solution:
problem_62 = undefined
Problem 63
How many n-digit positive integers exist which are also an nth power?
Solution:
problem_63 = undefined
Problem 64
How many continued fractions for N ≤ 10000 have an odd period?
Solution:
problem_64 = undefined
Problem 65
Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.
Solution:
problem_65 = undefined
Problem 66
Investigate the Diophantine equation x2 − Dy2 = 1.
Solution:
problem_66 = undefined
Problem 67
Using an efficient algorithm find the maximal sum in the triangle?
Solution:
problem_67 = undefined
Problem 68
What is the maximum 16-digit string for a "magic" 5-gon ring?
Solution:
problem_68 = undefined
Problem 69
Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
Solution:
problem_69 = undefined
Problem 70
Investigate values of n for which φ(n) is a permutation of n.
Solution:
problem_70 = undefined
Problem 71
Listing reduced proper fractions in ascending order of size.
Solution:
problem_71 = undefined
Problem 72
How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000?
Solution:
problem_72 = undefined
Problem 73
How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?
Solution:
problem_73 = undefined
Problem 74
Determine the number of factorial chains that contain exactly sixty non-repeating terms.
Solution:
problem_74 = undefined
Problem 75
Find the number of different lengths of wire can that can form a right angle triangle in only one way.
Solution:
problem_75 = undefined
Problem 76
How many different ways can one hundred be written as a sum of at least two positive integers?
Solution:
problem_76 = undefined
Problem 77
What is the first value which can be written as the sum of primes in over five thousand different ways?
Solution:
problem_77 = undefined
Problem 78
Investigating the number of ways in which coins can be separated into piles.
Solution:
problem_78 = undefined
Problem 79
By analysing a user's login attempts, can you determine the secret numeric passcode?
Solution:
problem_79 = undefined
Problem 80
Calculating the digital sum of the decimal digits of irrational square roots.
Solution:
problem_80 = undefined
Problem 81
Find the minimal path sum from the top left to the bottom right by moving right and down.
Solution:
problem_81 = undefined
Problem 82
Find the minimal path sum from the left column to the right column.
Solution:
problem_82 = undefined
Problem 83
Find the minimal path sum from the top left to the bottom right by moving left, right, up, and down.
Solution:
problem_83 = undefined
Problem 84
In the game, Monopoly, find the three most popular squares when using two 4-sided dice.
Solution:
problem_84 = undefined
Problem 85
Investigating the number of rectangles in a rectangular grid.
Solution:
problem_85 = undefined
Problem 86
Exploring the shortest path from one corner of a cuboid to another.
Solution:
problem_86 = undefined
Problem 87
Investigating numbers that can be expressed as the sum of a prime square, cube, and fourth power?
Solution:
problem_87 = undefined
Problem 88
Exploring minimal product-sum numbers for sets of different sizes.
Solution:
problem_88 = undefined
Problem 89
Develop a method to express Roman numerals in minimal form.
Solution:
problem_89 = undefined
Problem 90
An unexpected way of using two cubes to make a square.
Solution:
problem_90 = undefined
Problem 91
Find the number of right angle triangles in the quadrant.
Solution:
problem_91 = undefined
Problem 92
Investigating a square digits number chain with a surprising property.
Solution:
problem_92 = undefined
Problem 93
Using four distinct digits and the rules of arithmetic, find the longest sequence of target numbers.
Solution:
problem_93 = undefined
Problem 94
Investigating almost equilateral triangles with integral sides and area.
Solution:
problem_94 = undefined
Problem 95
Find the smallest member of the longest amicable chain with no element exceeding one million.
Solution:
problem_95 = undefined
Problem 96
Devise an algorithm for solving Su Doku puzzles.
Solution:
problem_96 = undefined
Problem 97
Find the last ten digits of the non-Mersenne prime: 28433 × 27830457 + 1.
Solution:
problem_97 = undefined
Problem 98
Investigating words, and their anagrams, which can represent square numbers.
Solution:
problem_98 = undefined
Problem 99
Which base/exponent pair in the file has the greatest numerical value?
Solution:
problem_99 = undefined
Problem 100
Finding the number of blue discs for which there is 50% chance of taking two blue.
Solution:
problem_100 = undefined
Problem 101
Investigate the optimum polynomial function to model the first k terms of a given sequence.
Solution:
problem_101 = undefined
Problem 102
For how many triangles in the text file does the interior contain the origin?
Solution:
problem_102 = undefined
Problem 103
Investigating sets with a special subset sum property.
Solution:
problem_103 = undefined
Problem 104
Finding Fibonacci numbers for which the first and last nine digits are pandigital.
Solution:
problem_104 = undefined
Problem 105
Find the sum of the special sum sets in the file.
Solution:
problem_105 = undefined
Problem 106
Find the minimum number of comparisons needed to identify special sum sets.
Solution:
problem_106 = undefined
Problem 107
Determining the most efficient way to connect the network.
Solution:
problem_107 = undefined
Problem 108
Solving the Diophantine equation 1/x + 1/y = 1/n.
Solution:
problem_108 = undefined
Problem 109
How many distinct ways can a player checkout in the game of darts with a score of less than 100?
Solution:
problem_109 = undefined
Problem 110
Find an efficient algorithm to analyse the number of solutions of the equation 1/x + 1/y = 1/n.
Solution:
problem_110 = undefined
Problem 111
Search for 10-digit primes containing the maximum number of repeated digits.
Solution:
problem_111 = undefined
Problem 112
Investigating the density of "bouncy" numbers.
Solution:
problem_112 = undefined
Problem 113
How many numbers below a googol (10100) are not "bouncy"?
Solution:
problem_113 = undefined
Problem 114
Investigating the number of ways to fill a row with separated blocks that are at least three units long.
Solution:
problem_114 = undefined
Problem 115
Finding a generalisation for the number of ways to fill a row with separated blocks.
Solution:
problem_115 = undefined
Problem 116
Investigating the number of ways of replacing square tiles with one of three coloured tiles.
Solution:
problem_116 = undefined
Problem 117
Investigating the number of ways of tiling a row using different-sized tiles.
Solution:
problem_117 = undefined
Problem 118
Exploring the number of ways in which sets containing prime elements can be made.
Solution:
problem_118 = undefined
Problem 119
Investigating the numbers which are equal to sum of their digits raised to some power.
Solution:
problem_119 = undefined
Problem 120
Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2.
Solution:
problem_120 = undefined
Problem 121
Investigate the game of chance involving coloured discs.
Solution:
problem_121 = undefined
Problem 122
Finding the most efficient exponentiation method.
Solution:
problem_122 = undefined
Problem 123
Determining the remainder when (pn − 1)n + (pn + 1)n is divided by pn2.
Solution:
problem_123 = undefined
Problem 124
Determining the kth element of the sorted radical function.
Solution:
problem_124 = undefined
Problem 125
Finding square sums that are palindromic.
Solution:
problem_125 = undefined
Problem 126
Exploring the number of cubes required to cover every visible face on a cuboid.
Solution:
problem_126 = undefined
Problem 127
Investigating the number of abc-hits below a given limit.
Solution:
problem_127 = undefined
Problem 128
Which tiles in the hexagonal arrangement have prime differences with neighbours?
Solution:
problem_128 = undefined
Problem 129
Investigating minimal repunits that divide by n.
Solution:
problem_129 = undefined
Problem 130
Finding composite values, n, for which n−1 is divisible by the length of the smallest repunits that divide it.
Solution:
problem_130 = undefined
Problem 131
Determining primes, p, for which n3 + n2p is a perfect cube.
Solution:
problem_131 = undefined
Problem 132
Determining the first forty prime factors of a very large repunit.
Solution:
problem_132 = undefined
Problem 133
Investigating which primes will never divide a repunit containing 10n digits.
Solution:
problem_133 = undefined
Problem 134
Finding the smallest positive integer related to any pair of consecutive primes.
Solution:
problem_134 = undefined
Problem 135
Determining the number of solutions of the equation x2 − y2 − z2 = n.
Solution:
problem_135 = undefined
Problem 136
Discover when the equation x2 − y2 − z2 = n has a unique solution.
Solution:
problem_136 = undefined
Problem 137
Determining the value of infinite polynomial series for which the coefficients are Fibonacci numbers.
Solution:
problem_137 = undefined
Problem 138
Investigating isosceles triangle for which the height and base length differ by one.
Solution:
problem_138 = undefined
Problem 139
Finding Pythagorean triangles which allow the square on the hypotenuse square to be tiled.
Solution:
problem_139 = undefined
Problem 140
Investigating the value of infinite polynomial series for which the coefficients are a linear second order recurrence relation.
Solution:
problem_140 = undefined
Problem 141
Investigating progressive numbers, n, which are also square.
Solution:
problem_141 = undefined
Problem 142
Perfect Square Collection
Solution:
problem_142 = undefined
Problem 143
Investigating the Torricelli point of a triangle
Solution:
problem_143 = undefined
Problem 144
Investigating multiple reflections of a laser beam.
Solution:
problem_144 = undefined
Problem 145
How many reversible numbers are there below one-billion?
Solution:
problem_145 = undefined
Problem 146
Investigating a Prime Pattern
Solution:
problem_146 = undefined