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| Numbered Words (Posted on 2011-11-04) |
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Suppose A=1, B=2, C=3, ..., Z=26. Then, ETUDE is an interesting word. It has 5 letters, it starts and ends with the 5th letter of the alphabet, and E+T+U+D+E=5+20+21+4+5=55. What other words both start and end with the Xth letter of the alphabet and add up to XX, where X is the number of letters in the word and XX is X concatenated with X? (For ETUDE, X=5.)
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Submitted by Math Man
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Rating: 4.0000 (1 votes)
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Solution:
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It is easy to see that the word cannot have 1 or 2 letters. If it has 3 letters, then the middle letter has to have a value of 27 to add up to 33. That is impossible. If it has 10 or more letters, then it cannot add up to XX. Therefore, it must have between 4 and 9 letters. Here are the words with this property.
5 letters:
etude
E+T+U+D+E=5+20+21+4+5=55
exile
E+X+I+L+E=5+24+9+12+5=55
7 letters:
gorging
G+O+R+G+I+N+G=7+15+18+7+9+14+7=77
griming
G+R+I+M+I+N+G=7+18+9+13+9+14+7=77
8 letters:
hyacinth
H+Y+A+C+I+N+T+H=8+25+1+3+9+14+20+8=88
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