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Problem 42

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Coded triangle numbers

The $n$^th term of the sequence of triangle numbers is given by, $t_n=n (n+1)/2$; so the first ten triangle numbers are:

$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots$$

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?

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编码三角形数

三角形数序列的第$n$项由公式$t_n=n (n+1)/2$给出。前十个三角形数是：

$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots$$

将一个单词的每个字母分别转化为其在字母表中的顺序并相加，所得结果即为这个单词的价值。例如，单词SKY的价值是$19 + 11 + 25 = 55 = t_{10}$。如果一个单词的价值是一个三角形数，称这个单词为三角形单词。

在文本文件words.txt中包含有将近两千个常用英文单词，其中有多少个三角形单词？

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