Many centuries ago, in ancient Alexandria, an old man had to bury his son. Heartbroken, the man distracted himself by assembling a large collection of algebra problems and their solutions in a book he called the Arithmetica. That is practically all that is known of Diophantus of Alexandria, and most of it comes from a riddle believed to have been written by a close friend soon after his death1:
This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. 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.2
The epitaph is a bit ambiguous regarding the death of Diophantus's son. He is said to have died at “half his father's life,” but does that mean half the father's age at the time of the son's death, or half the age at which Diophantus himself eventually died? You can work it out either way, but the latter assumption — Diophantus's son lived half the number of years that Diophantus eventually did — is the one with the nice, clean solution in whole numbers without fractional years.
Let's represent ...