## Project Euler 42: Count how many 'triangle words' a list of common English words contain

#### Problem Description

The *n*^{th} term of the sequence of triangle numbers is given by, *t _{n}* = ½

*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?

#### Analysis

Remember Problems 44 & 45? The same logic applies except for triangle numbers. Whenever the equation ( √(1+8n) - 1 ) / 2 evaluates to an integer the number, *n*, is a triangle number. The *mod 1* trick is a quick way to tell if the expression resolved as an integer.

In a nutshell: We connect to the file, strip quotes and separate by commas into a list. We define two functions: one to check a number for triangularity and the other to calculate a word score. Finally, we count the number of scores that are triangular.

#### Project Euler 42 Solution

Runs < 0.001 seconds in Python 2.7.```
from math import sqrt
from urllib2 import urlopen
file_url = 'http://projecteuler.net/project/words.txt'
words = [x[1:-1] for x in urlopen(file_url).read().split(',')]
def is_tn(n): return not bool(((sqrt(1 + 8*n) - 1) / 2) % 1)
def score(word): return sum(abs(64 - ord(letter)) for letter in word)
print "Project Euler 42 Solution =", sum(is_tn(score(word)) for word in words)
```

Use this link to get the Project Euler 42 Solution Python 2.7 source.#### Answer

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