# Problem 42

The nth term of the sequence of triangle numbers is given by, tn = ½n(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 = t10. 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?

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In [2]:

_alphabet_value = {}
for i in list('ABCDEFGHIJKLMNOPQRSTUVWXYZ'):
_alphabet_value[i] = len(_alphabet_value)+1

def alphabet_value(alphabet):
return _alphabet_value[alphabet]

def word_value(word):
return sum([alphabet_value(i) for i in list(word.upper())])

def calculate_triangle_number(n):
return n * (n+1) / 2

_triangle_numbers = set([1])
def is_triangle_number(t):
while max(_triangle_numbers) < t:
n = len(_triangle_numbers) + 1
_triangle_numbers.add(calculate_triangle_number(n))
return t in _triangle_numbers

def count_triangle_words(word_list):
count = 0
for word in word_list:
if is_triangle_number(word_value(word)):
count += 1
return count

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In [3]:

import time
import requests
response = requests.get('https://projecteuler.net/project/resources/p042_words.txt')
words = map(lambda w: w[1:-1], response.text.split(','))
t0 = time.time()
ans = count_triangle_words(words)
t1 = time.time()
print('Answer:', ans)
print('Elapsed:', t1-t0)

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Answer: 162
Elapsed: 0.017287015914916992

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In [ ]:

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