## Problem

The $n^\text{th}$ term of the sequence of triangle numbers is given by, $t_n=\frac{1}{2}n(n+1)$; so the first ten triangle numbers are: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …

By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is 19 + 11 + 25 = 55 = $t_{10}$. If the word value is a triangle number then we shall call the word a triangle word.

Using words.txt (right click and ‘Save Link/Target As…’), a 16K text file containing nearly two-thousand common English words, how many are triangle words?

## Solution in Python

#!/usr/bin/python   import time   # start the clock start = time.time()   # turn the string of words into a list of python strings with open('words.txt','r') as f: words = f.read().split(',') words = [list(word.strip('\"')) for word in words] f.close()   # we should have an idea of how long the longest word is, # giving us an idea of the magnitude of the triangle words m = max([len(word) for word in words]) # form triangle numbers up to the given range triangles = [n*(n + 1)/2 for n in range(1,2*m)]   # make a dictionary map for character values vals = {} s = list("ABCDEFGHIJKLMNOPQRSTUVWXYZ") for c in s: vals[c] = s.index(c) + 1   # count triangle words triangle_words = 0 for word in words: if sum([vals[c] for c in word]) in triangles: triangle_words += 1   # terminate the clock elapsed = time.time() - start   print "found %s triangle words in %s seconds" % (triangle_words, elapsed)

When executed, this returns as follows.

found 162 triangle words in 0.00315809249878 seconds