The problem description is here, and click here to see all my other Euler solutions in F#. This is a more difficult version of problem 82, and now you can move in all four directions! As before, we start by loading the input data into a 2D array: and initialize another matrix of the same …
The problem description is here, and click here to see all my other Euler solutions in F#. This is a more difficult version of problem 81, but still, as you can’t move left so we can still optimize one column at a time. First, let’s read the input file into a 2D array: and …
The problem description is here, and click here to see all my other Euler solutions in F#. After reading the question, a quick search on how to test if a point is in a triangle turned up this useful SO answer. Translating the algorithm to F# is pretty trivial: I saved the triangle.txt file …
The problem description is here, and click here to see all my other Euler solutions in F#. I based my solution on Euclid’s formula for generating Pythagorean triples. And given that max L is 1,500,000, the maximum value for m we need to consider is $latex \sqrt{\frac{L}{2}} $. Because $latex L = a + b + …
Problem Consider the following “magic” 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine. Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described by the …
Problem All square roots are periodic when written as continued fractions and can be written in the form: For example, let us consider ?23: If we continue we would get the following expansion: The process can be summarised as follows: It can be seen that the sequence is repeating. For conciseness, we use the notation …
Problem It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all. The square root of two is 1.41421356237309504880…, and the digital sum of the first one hundred decimal digits …
Problem Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae: The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties. The set is cyclic, in that the last two digits of each number is the first two digits of the …
Problem The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes, 792, represents the lowest sum for a set of …
Problem The square root of 2 can be written as an infinite continued fraction. The infinite continued fraction can be written, ?2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, ?23 = [4;(1,3,1,8)]. It turns out that the sequence of partial values of continued fractions for square roots provide the …
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