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.)

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.