Problem 61

Projekt Euler w Ruby
1

Problem 61

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
Triangle P[sub]3,n[/sub]=n(n+1)/2 1, 3, 6, 10, 15, …
Square P[sub]4,n[/sub]=n[sup]2[/sup] 1, 4, 9, 16, 25, …
Pentagonal P[sub]5,n[/sub]=n(3n−1)/2 1, 5, 12, 22, 35, …
Hexagonal P[sub]6,n[/sub]=n(2n−1) 1, 6, 15, 28, 45, …
Heptagonal P[sub]7,n[/sub]=n(5n−3)/2 1, 7, 18, 34, 55, …
Octagonal P[sub]8,n[/sub]=n(3n−2) 1, 8, 21, 40, 65, …

The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.

  1. The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
  2. Each polygonal type: triangle (P[sub]3,127[/sub]=8128), square (P[sub]4,91[/sub]=8281), and pentagonal (P[sub]5,44[/sub]=2882), is represented by a different number in the set.
  3. This is the only set of 4-digit numbers with this property.

Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.

2

[code=ruby]#!/usr/bin/env ruby

def cycles(th, ti, x, h)
l = x%100
unless ti == Array.new(ti.length)
tz = nil
ti.detect do |i|
if i
tx = th[i][l].detect do |y|
tj = ti.dup
tj[i] = nil
ty = cycles(th, tj, y, h)
tz = [x]+ty if ty
end
end
end
tz
else
[x] if l == h
end
end

tf = [lambda { |n| n*(n+1)/2 },
lambda { |n| n**2 },
lambda { |n| n*(3n-1)/2 },
lambda { |n| n(2n-1) },
lambda { |n| n(5n-3)/2 },
lambda { |n| n(3*n-2) }]

th = Array.new(tf.length) { Array.new(100) { [] } }
first = []

0.upto(tf.length-1) do |i|
f = tf[i]
n = 1
while (x = f.call(n)) < 10000
if x >= 1000
th[i][x/100] << x
first << x if i == 0
end
n += 1
end
end

ty = nil
first.detect do |x|
tx = cycles(th, [nil]+(1…th.length-1).to_a, x, x/100)
ty = tx if tx
end

if ty
sum = ty.inject(0) { |s, x| s+x }
puts “cycles = [#{ty.join(’, ')}], sum = #{sum}”
else
puts “Sorry”
end[/code]