The well-known riddle of the Epitaph of Diophantus, attributed to Metrodorus, is one of the style of simple problem in algebra whose pattern when expressed in contemporary algebraic notation is:

$$x = \dfrac x 6 + \dfrac x {12} + \dfrac x 7 + 5 + \dfrac x 2 + 4$$

This appears in the Greek Anthology Book XIV as epigram no. 126.

The version that appears in the W.R. Paton translation is best known:

This tomb holds Diophantus. Ah, how great a marvel! the tomb tells scientifically the measure of his life. God granted him to be a boy for the sixth part of his life, and adding a twelfth part to this, he clothed his cheeks with down; He lit him the light of wedlock after a seventh part, and five years after his marriage He granted him a son. Alas! late-born wretched child; after attaining the measure of half his father's life, chill Fate took him. After consoling his grief by this science of numbers for four years he ended his life.

Now a large swathe of the problems in the Greek Anthology are along similar lines, not only in the context of working out someone's age (of which there are quite a few) but in a whole raft of different contexts:

  • how many students attended Pythagoras's Academy (epigram 1)
  • the portions of gold donated to the statue of Pallas (epigram 2)
  • how many apples were stolen by the Muses from Cupid (epigram 3)
  • how many herds of cattle Augeas owned (epigram 4)
  • how many walnuts were stolen during a highway mugging by a posse of pretty girls (epigram 116)
  • how many apples were stolen during a similar robbery (epigram 117)
  • a more civilised distribution of apples (epigram 118)
  • the number of walnuts harvested from a tree (epigram 120)
  • how far it is from Cadiz to Rome (epigram 121)
  • how large a fortune was squandered (epigram 122)
  • how an inheritance was distributed (epigram 123)
  • the age of an unnamed protagonist (epigram 124)
  • how many children were born to Philinna (epigram 125)
  • epitaph of Demochares (epigram 127)
  • how many people were killed when Antiochus's house collapsed (epigram 137)
  • how Nicarete distributed her nuts (epigram 138)

Clearly these all fit the same pattern: you are given a description of how a number is broken down into a set of both fractions of the whole and one or more specific enumerations, and you are asked to calculate what that number is.

Such a puzzle can be seen in a number of such collections of puzzles (if I'm not mistaken Alcuin includes some in his Propositiones ad Acuendos Juvenes), but only the Epitaph of Diophantus seems to be well-known.

My question is: is there an accepted name for this classification of problem?

  • 1
    $\begingroup$ These are the original "Diophantine equations". $\endgroup$ Feb 8, 2022 at 11:58
  • $\begingroup$ @GeraldEdgar Well fair enough, but you can't really call such a problem a "Diophantine equation" because the latter term has a somewhat different scope than just this specific type of fractions-plus-specific-numbers-equal-the-whole problem. $\endgroup$ Feb 8, 2022 at 16:52


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