### Brain Teasers

# Word on the Stone

The archaeologist Doug Wither-Trowel was excavating a site in France when he came upon a four-letter English word engraved on a stone. In his notebook, he wrote that no two letters were identical, if one replaced each letter in the word with a number giving its alphabetic position (A=1, B=2 etc) the total of the letters is 20, and the sum of any three of the numbers is exactly divisible by the fourth number. His French assistant, reading the entry in the notebook, claimed the word was French. What was the English word on the stone?

### Answer

The English word was JADE (the French was DEJA). If the four numbers are W, X, Y and Z and their total is T then(W + X + Y)/Z = (T - Z)/Z

Since the left-hand side of the equation is a whole number then both (T/Z) - 1 and T/Z are also whole numbers. By symmetry, W, X, Y, Z are factors of T. The possible factors of 20 are 1, 2, 4, 5, 10, 20. The four of these that sum to 20 are 1, 4, 5, 10. Replacing these by letters gives A, D, J, E.

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