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?



In [1]:

    
f = open('./p042_words.txt', 'r')
t = f.read()
f.close()

t = [a[1:-1] for a in t.split(',')]
print len(t)



In [2]:

    
import math

# Since n*n <= 2*t < n*(n+1), n = floor(sqrt(2*t)) needs
# to be divisible by (n+1) to be triangular.
def is_triangular(w):
    s = 2*sum([ord(c) - 64 for c in w])
    n = int(math.floor(math.sqrt(s)))
    
    return s % (n + 1) == 0
assert is_triangular('SKY') == True 

# Alternatively, can use quadratic formula and then
# check if result is an integer.
def is_triangular_alternative(w):
    s = sum([ord(c) - 64 for c in w])
    n = (-1  + math.sqrt(1 + 8*s))/2.0
    
    return abs(n - math.floor(n)) < 10e-6
assert is_triangular_alternative('SKY') == True



In [3]:

    
c = 0
for w in t:
    if is_triangular(w):
        c += 1
print c