Project Euler

Project Euler is a website that lists math/programming problems to be solved for fun, or to help learn a new programming language. I recently started doing some of the problems to aid me in learning Clojure, and decided it would be interesting to compile my solutions. The table below lists all of the problems on Project Euler, and links to the problem description and my solutions. On the right are columns listing the running time for my solutions in each language. The problems are designed to be solvable in less than a minute in program running time, so any of my solutions that run longer than that are marked in red.

#	Problem	Clojure	Erlang	Ruby	Scala
1	Add all the natural numbers below one thousand that are multiples of 3 or 5.	1.54	0.198	0.01	0.24
2	Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed four million.	1.61	0.417	0.02	0.21
3	Find the largest prime factor of a composite number.	-	0.14	0.03	0.26
4	Find the largest palindrome made from the product of two 3-digit numbers.	6.04	0.52	3.64	1.35
5	What is the smallest number divisible by each of the numbers 1 to 20?	-	0.12	0.02	-
6	What is the difference between the sum of the squares and the square of the sums?	1.76	0.14	0.02	0.18
7	Find the 10001<sup>st</sup> prime.	-	0.56	0.98	0.51
8	Discover the largest product of five consecutive digits in the 1000-digit number.	-	0.12	0.05	0.33
9	Find the only Pythagorean triplet, {<var>a</var>, <var>b</var>, <var>c</var>}, for which <var>a</var> + <var>b</var> + <var>c</var> = 1000.	-	0.473	0.338	0.5
10	Calculate the sum of all the primes below two million.	-	10.45	28.48	0.86
11	What is the greatest product of four numbers on the same straight line in the 20 by 20 grid?	-	0.12	0.0	0.25
12	What is the value of the first triangle number to have over five hundred divisors?	-	9.57	4.56	0.52
13	Find the first ten digits of the sum of one-hundred 50-digit numbers.	-	0.433	0.02	-
14	Find the longest sequence using a starting number under one million.	-	7.133	13.64	-
15	Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner?	-	0.11	0.02	-
16	What is the sum of the digits of the number 2<sup>1000</sup>?	1.56	0.12	0.02	0.22
17	How many letters would be needed to write all the numbers in words from 1 to 1000?	-	0.203	0.04	-
18	Find the maximum sum travelling from the top of the triangle to the base.	-	0.21	0.02	-
19	How many Sundays fell on the first of the month during the twentieth century?	-	0.205	0.021	-
20	Find the sum of digits in 100!	1.62	0.12	0.01	0.19
21	Evaluate the sum of all amicable pairs under 10000.	-	4.66	3.51	-
22	What is the total of all the name scores in the file of first names?	-	0.278	0.12	-
23	Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.	-	18.0	9.45	-
24	What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?	-	0.08	0.0	-
25	What is the first term in the Fibonacci sequence to contain 1000 digits?	3.58	0.13	0.032	-
26	Find the value of <i>d</i> < 1000 for which 1/<i>d</i> contains the longest recurring cycle.	-	-	-	-
27	Find a quadratic formula that produces the maximum number of primes for consecutive values of <i>n</i>.	-	4.62	7.23	-
28	What is the sum of both diagonals in a 1001 by 1001 spiral?	-	0.204	0.03	-
29	How many distinct terms are in the sequence generated by <i>a<sup>b</sup></i> for 2 ≤ <i>a</i> ≤ 100 and 2 ≤ <i>b</i> ≤ 100?	3.37	0.2	0.08	-
30	Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.	-	1.56	6.82	-
31	Investigating combinations of English currency denominations.	-	7.165	0.372	-
32	Find the sum of all numbers that can be written as pandigital products.	-	3.58	0.98	-
33	Discover all the fractions with an unorthodox cancelling method.	-	-	0.12	-
34	Find the sum of all numbers which are equal to the sum of the factorial of their digits.	-	-	0.78	-
35	How many circular primes are there below one million?	-	9.1	6.46	-
36	Find the sum of all numbers less than one million, which are palindromic in base 10 and base 2.	5.29	0.62	2.06	-
37	Find the sum of all eleven primes that are both truncatable from left to right and right to left.	-	-	0.1	-
38	What is the largest 1 to 9 pandigital that can be formed by multiplying a fixed number by 1, 2, 3, ... ?	-	-	0.221	-
39	If <i>p</i> is the perimeter of a right angle triangle, {<i>a</i>, <i>b</i>, <i>c</i>}, which value, for <i>p</i> ≤ 1000, has the most solutions?	-	-	0.471	-
40	Finding the <i>n</i><sup>th</sup> digit of the fractional part of the irrational number.	-	-	0.438	-
41	What is the largest <i>n</i>-digit pandigital prime that exists?	-	-	1.09	-
42	How many triangle words does the list of common English words contain?	-	0.13	0.05	-
43	Find the sum of all pandigital numbers with an unusual sub-string divisibility property.	-	0.1	0.02	-
44	Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.	-	-	45.97	-
45	After 40755, what is the next triangle number that is also pentagonal and hexagonal?	-	-	2.05	-
46	What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?	-	-	0.93	-
47	Find the first four consecutive integers to have four distinct primes factors.	-	37.73	19.79	-
48	Find the last ten digits of 1<sup>1</sup> + 2<sup>2</sup> + ... + 1000<sup>1000</sup>.	2.08	0.13	0.0	-
49	Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other.	-	-	0.0	-
50	Which prime, below one-million, can be written as the sum of the most consecutive primes?	-	8.65	0.67	-
51	Find the smallest prime which, by changing the same part of the number, can form eight different primes.	-	-	-	-
52	Find the smallest positive integer, <i>x</i>, such that 2<i>x</i>, 3<i>x</i>, 4<i>x</i>, 5<i>x</i>, and 6<i>x</i>, contain the same digits in some order.	-	2.79	4.32	-
53	How many values of C(<i>n</i>,<i>r</i>), for 1 ≤ <i>n</i> ≤ 100, exceed one-million?	-	0.19	0.25	-
54	How many hands did player one win in the game of poker?	-	-	0.13	-
55	How many Lychrel numbers are there below ten-thousand?	-	0.54	0.2	-
56	Considering natural numbers of the form, <i>a<sup>b</sup></i>, finding the maximum digital sum.	5.47	0.38	2.18	-
57	Investigate the expansion of the continued fraction for the square root of two.	-	-	3.66	-
58	Investigate the number of primes that lie on the diagonals of the spiral grid.	-	9.78	42.71	-
59	Using a brute force attack, can you decrypt the cipher using XOR encryption?	-	-	-	-
60	Find a set of five primes for which any two primes concatenate to produce another prime.	-	-	-	-
61	Find the sum of the only set of six 4-digit figurate numbers with a cyclic property.	-	-	1.28	-
62	Find the smallest cube for which exactly five permutations of its digits are cube.	-	0.2	0.31	-
63	How many <i>n</i>-digit positive integers exist which are also an <i>n</i><sup>th</sup> power?	-	-	0.023	-
64	How many continued fractions for <i>N</i> ≤ 10000 have an odd period?	-	-	-	-
65	Find the sum of digits in the numerator of the 100<sup>th</sup> convergent of the continued fraction for <i>e</i>.	-	-	0.04	-
66	Investigate the Diophantine equation <i>x</i><sup>2</sup> − D<i>y</i><sup>2</sup> = 1.	-	-	-	-
67	Using an efficient algorithm find the maximal sum in the triangle?	-	0.476	0.061	-
68	What is the maximum 16-digit string for a "magic" 5-gon ring?	-	-	-	-
69	Find the value of <i>n</i> ≤ 1,000,000 for which <i>n</i>/φ(<i>n</i>) is a maximum.	-	0.14	-	-
70	Investigate values of <var>n</var> for which φ(<var>n</var>) is a permutation of <var>n</var>.	-	-	-	-
71	Listing reduced proper fractions in ascending order of size.	-	-	41.44	-
72	How many elements would be contained in the set of reduced proper fractions for <i>d</i> ≤ 1,000,000?	-	28.71	-	-
73	How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?	-	-	47.35	-
74	Determine the number of factorial chains that contain exactly sixty non-repeating terms.	-	12.94	-	-
75	Find the number of different lengths of wire can that can form a right angle triangle in only one way.	-	-	-	-
76	How many different ways can one hundred be written as a sum of at least two positive integers?	-	-	-	-
77	What is the first value which can be written as the sum of primes in over five thousand different ways?	-	-	-	-
78	Investigating the number of ways in which coins can be separated into piles.	-	-	-	-
79	By analysing a user's login attempts, can you determine the secret numeric passcode?	-	-	-	-
80	Calculating the digital sum of the decimal digits of irrational square roots.	-	-	2.4	-
81	Find the minimal path sum from the top left to the bottom right by moving right and down.	-	-	-	-
82	Find the minimal path sum from the left column to the right column.	-	-	-	-
83	Find the minimal path sum from the top left to the bottom right by moving left, right, up, and down.	-	-	-	-
84	In the game, Monopoly, find the three most popular squares when using two 4-sided dice.	-	-	-	-
85	Investigating the number of rectangles in a rectangular grid.	-	-	-	-
86	Exploring the shortest path from one corner of a cuboid to another.	-	-	-	-
87	Investigating numbers that can be expressed as the sum of a prime square, cube, and fourth power?	-	4.08	-	-
88	Exploring minimal product-sum numbers for sets of different sizes.	-	-	-	-
89	Develop a method to express Roman numerals in minimal form.	-	-	-	-
90	An unexpected way of using two cubes to make a square.	-	-	-	-
91	Find the number of right angle triangles in the quadrant.	-	23.664	-	-
92	Investigating a square digits number chain with a surprising property.	-	-	0.76	-
93	Using four distinct digits and the rules of arithmetic, find the longest sequence of target numbers.	-	-	-	-
94	Investigating almost equilateral triangles with integral sides and area.	-	-	-	-
95	Find the smallest member of the longest amicable chain with no element exceeding one million.	-	-	-	-
96	Devise an algorithm for solving Su Doku puzzles.	-	-	2.49	-
97	Find the last ten digits of the non-Mersenne prime: 28433 × 2<sup>7830457</sup> + 1.	1.63	0.241	0.02	-
98	Investigating words, and their anagrams, which can represent square numbers.	-	-	-	-
99	Which base/exponent pair in the file has the greatest numerical value?	-	0.222	0.035	-
100	Finding the number of blue discs for which there is 50% chance of taking two blue.	-	-	-	-
101	Investigate the optimum polynomial function to model the first <i>k</i> terms of a given sequence.	-	-	-	-
102	For how many triangles in the text file does the interior contain the origin?	-	-	-	-
103	Investigating sets with a special subset sum property.	-	-	-	-
104	Finding Fibonacci numbers for which the first and last nine digits are pandigital.	-	67.756	-	-
105	Find the sum of the special sum sets in the file.	-	-	-	-
106	Find the minimum number of comparisons needed to identify special sum sets.	-	-	-	-
107	Determining the most efficient way to connect the network.	-	-	-	-
108	Solving the Diophantine equation 1/<var>x</var> + 1/<var>y</var> = 1/<var>n</var>.	-	-	-	-
109	How many distinct ways can a player checkout in the game of darts with a score of less than 100?	-	-	-	-
110	Find an efficient algorithm to analyse the number of solutions of the equation 1/<var>x</var> + 1/<var>y</var> = 1/<var>n</var>.	-	-	-	-
111	Search for 10-digit primes containing the maximum number of repeated digits.	-	-	-	-
112	Investigating the density of "bouncy" numbers.	-	8.078	-	-
113	How many numbers below a googol (10<sup>100</sup>) are not "bouncy"?	-	-	-	-
114	Investigating the number of ways to fill a row with separated blocks that are at least three units long.	-	-	-	-
115	Finding a generalisation for the number of ways to fill a row with separated blocks.	-	-	-	-
116	Investigating the number of ways of replacing square tiles with one of three coloured tiles.	-	-	0.039	-
117	Investigating the number of ways of tiling a row using different-sized tiles.	-	-	0.031	-
118	Exploring the number of ways in which sets containing prime elements can be made.	-	-	-	-
119	Investigating the numbers which are equal to sum of their digits raised to some power.	-	-	-	-
120	Finding the maximum remainder when (<i>a</i> − 1)<sup><i>n</i></sup> + (<i>a</i> + 1)<sup><i>n</i></sup> is divided by <i>a</i><sup>2</sup>.	-	-	-	-
121	Investigate the game of chance involving coloured discs.	-	-	-	-
122	Finding the most efficient exponentiation method.	-	-	-	-
123	Determining the remainder when (<i>p<sub>n</sub></i> − 1)<sup><i>n</i></sup> + (<i>p<sub>n</sub></i> + 1)<sup><i>n</i></sup> is divided by <i>p<sub>n</sub></i><sup>2</sup>.	-	-	-	-
124	Determining the <i>k</i><sup>th</sup> element of the sorted radical function.	-	-	15.94	-
125	Finding square sums that are palindromic.	-	-	1.24	-
126	Exploring the number of cubes required to cover every visible face on a cuboid.	-	-	-	-
127	Investigating the number of <i>abc-hits</i> below a given limit.	-	-	-	-
128	Which tiles in the hexagonal arrangement have prime differences with neighbours?	-	-	-	-
129	Investigating minimal repunits that divide by <i>n</i>.	-	-	-	-
130	Finding composite values, <i>n</i>, for which <i>n</i>−1 is divisible by the length of the smallest repunits that divide it.	-	-	-	-
131	Determining primes, <i>p</i>, for which <i>n</i><sup>3</sup> + <i>n</i><sup>2</sup><i>p</i> is a perfect cube.	-	-	-	-
132	Determining the first forty prime factors of a very large repunit.	-	-	-	-
133	Investigating which primes will never divide a repunit containing 10<sup><i>n</i></sup> digits.	-	-	-	-
134	Finding the smallest positive integer related to any pair of consecutive primes.	-	-	-	-
135	Determining the number of solutions of the equation <i>x</i><sup>2</sup> − <i>y</i><sup>2</sup> − <i>z</i><sup>2</sup> = <i>n</i>.	-	-	-	-
136	Discover when the equation <i>x</i><sup>2</sup> − <i>y</i><sup>2</sup> − <i>z</i><sup>2</sup> = <i>n</i> has a unique solution.	-	-	-	-
137	Determining the value of infinite polynomial series for which the coefficients are Fibonacci numbers.	-	-	-	-
138	Investigating isosceles triangle for which the height and base length differ by one.	-	-	-	-
139	Finding Pythagorean triangles which allow the square on the hypotenuse square to be tiled.	-	-	-	-
140	Investigating the value of infinite polynomial series for which the coefficients are a linear second order recurrence relation.	-	-	-	-
141	Investigating progressive numbers, <i>n</i>, which are also square.	-	-	-	-
142	Perfect Square Collection	-	-	-	-
143	Investigating the Torricelli point of a triangle	-	-	-	-
144	Investigating multiple reflections of a laser beam.	-	-	-	-
145	How many reversible numbers are there below one-billion?	-	-	-	-
146	Investigating a Prime Pattern	-	-	-	-
147	Rectangles in cross-hatched grids	-	-	-	-
148	Exploring Pascal's triangle.	-	-	-	-
149	Searching for a maximum-sum subsequence.	-	-	-	-
150	Searching a triangular array for a sub-triangle having minimum-sum.	-	-	-	-
151	Paper sheets of standard sizes: an expected-value problem.	-	-	-	-
152	Writing 1/2 as a sum of inverse squares	-	-	-	-
153	Investigating Gaussian Integers	-	-	-	-
154	Exploring Pascal's pyramid.	-	-	-	-
155	Counting Capacitor Circuits.	-	-	-	-
156	Counting Digits	-	-	-	-
157	Solving the diophantine equation <sup>1</sup>/<sub><var>a</var></sub>+<sup>1</sup>/<sub><var>b</var></sub>= <sup><var>p</var></sup>/<sub>10<sup><var>n</var></sup></sub>	-	-	-	-
158	Exploring strings for which only one character comes lexicographically after its neighbour to the left.	-	-	-	-
159	Digital root sums of factorisations.	-	-	-	-
160	Factorial trailing digits	-	-	-	-
161	Triominoes	-	-	-	-
162	Hexadecimal numbers	-	0.428	0.02	-
163	Cross-hatched triangles	-	-	-	-
164	Numbers for which no three consecutive digits have a sum greater than a given value.	-	-	-	-
165	Intersections	-	-	-	-
166	Criss Cross	-	-	-	-
167	Investigating Ulam sequences	-	-	-	-
168	Number Rotations	-	-	-	-
169	Exploring the number of different ways a number can be expressed as a sum of powers of 2.	-	-	-	-
170	Find the largest 0 to 9 pandigital that can be formed by concatenating products.	-	-	-	-
171	Finding numbers for which the sum of the squares of the digits is a square.	-	-	-	-
172	Investigating numbers with few repeated digits.	-	-	-	-
173	Using up to one million tiles how many different "hollow" square laminae can be formed?	-	-	-	-
174	Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements.	-	-	-	-
175	Fractions involving the number of different ways a number can be expressed as a sum of powers of 2.	-	-	-	-
176	Rectangular triangles that share a cathetus.	-	-	-	-
177	Integer angled Quadrilaterals.	-	-	-	-
178	Step Numbers	-	-	-	-
179	Consecutive positive divisors	-	-	-	-
180	Rational zeros of a function of three variables.	-	-	-	-
181	Investigating in how many ways objects of two different colours can be grouped.	-	-	-	-
182	RSA encryption	-	-	-	-
183	Maximum product of parts	-	-	-	-
184	Triangles containing the origin.	-	-	-	-
185	Number Mind	-	-	-	-
186	Connectedness of a network.	-	-	-	-
187	Semiprimes	-	-	-	-
188	The hyperexponentiation of a number	-	-	-	-
189	Tri-colouring a triangular grid	-	-	-	-
190	Maximising a weighted product	-	-	-	-
191	Prize Strings	-	-	-	-
192	Best Approximations	-	-	-	-
193	Squarefree Numbers	-	-	-	-
194	Coloured Configurations	-	-	-	-
195	Inscribed circles of triangles with one angle of 60 degrees	-	-	-	-
196	Prime triplets	-	-	-	-
197	Investigating the behaviour of a recursively defined sequence	-	-	-	-
198	Ambiguous Numbers	-	-	-	-
199	Iterative Circle Packing	-	-	-	-
200	Find the 200th prime-proof sqube containing the contiguous sub-string "200"	-	-	-	-
201	Subsets with a unique sum	-	-	-	-
202	Laserbeam	-	-	-	-
203	Squarefree Binomial Coefficients	-	-	-	-
204	Generalised Hamming Numbers	-	4.476	-	-
205	Dice Game	-	2.722	-	-
206	Concealed Square	-	-	-	-
207	Integer partition equations	-	-	-	-
208	Robot Walks	-	-	-	-
209	Circular Logic	-	-	-	-
210	Obtuse Angled Triangles	-	-	-	-
211	Divisor Square Sum	-	-	-	-
212	Combined Volume of Cuboids	-	-	-	-
213	Flea Circus	-	-	-	-
214	Totient Chains	-	-	-	-
215	Crack-free Walls	-	-	-	-
216	Investigating the primality of numbers of the form 2<var>n</var><sup>2</sup>-1	-	-	-	-
217	Balanced Numbers	-	-	-	-
218	Perfect right-angled triangles	-	-	-	-
219	Skew-cost coding	-	-	-	-
220	Heighway Dragon	-	-	-	-
221	Alexandrian Integers	-	-	-	-
222	Sphere Packing	-	-	-	-
223	Almost right-angled triangles I	-	-	-	-
224	Almost right-angled triangles II	-	-	-	-
225	Tribonacci non-divisors	-	-	-	-
226	A Scoop of Blancmange	-	-	-	-
227	The Chase	-	-	-	-
228	Minkowski Sums	-	-	-	-
229	Four Representations using Squares	-	-	-	-
230	Fibonacci Words	-	-	-	-
231	The prime factorisation of binomial coefficients	-	-	-	-
232	The Race	-	-	-	-
233	Lattice points on a circle	-	-	-	-
234	Semidivisible numbers	-	-	-	-
235	An Arithmetic Geometric sequence	-	-	-	-
236	Luxury Hampers	-	-	-	-
237	Tours on a 4 x n playing board	-	-	-	-
238	Infinite string tour	-	-	-	-
239	Twenty-two Foolish Primes	-	-	-	-
240	Top Dice	-	-	-	-
241	Perfection Quotients	-	-	-	-
242	Odd Triplets	-	-	-	-
243	Resilience	-	-	-	-
244	Sliders	-	-	-	-
245	Coresilience	-	-	-	-
246	Tangents to an ellipse	-	-	-	-
247	Squares under a hyperbola	-	-	-	-
248	Numbers for which Euler’s totient function equals 13!	-	-	-	-
249	Prime Subset Sums	-	-	-	-
250	250250	-	-	-	-
251	Cardano Triplets	-	-	-	-
252	Convex Holes	-	-	-	-