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# Diophantus' riddle

Diophantus' riddle is one of the oldest known age puzzles. It is included in a collection of puzzles and epigrams compiled by the Greek mathematician and grammarian Metrodorus, and purports to tell how long Diophantus lived in the form of a riddle engraved on his tombstone:

God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.

If d and s are the ages of Diophantus and his son when they died, then the epitaph boils down to these two equations:

d = (1/6 + 1/12 + 1/7)d + 5 + s + 4
s = 1/2 d

which can be solved simultaneously to give s = 42 years and d = 84 years.