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Diophantus' Riddle

``Diophantus' youth lasts 1/6 of his life. He grew a beard after 1/12 more of his life. After 1/7 more of his life, Diophantus married. Five years later, he had a son. The son lived exactly half as long as his father, and Diophantus died just four years after his son's death. All of this totals the years Diophantus lived.''

Let $D$ be the number of years Diophantus lived, and let $S$ be the number of years his son lived. Then the above word problem gives the two equations

$\displaystyle D$ $\textstyle =$ $\displaystyle ({\textstyle{1\over 6}}+{\textstyle{1\over 12}}+{\textstyle{1\over 7}})D+5+S+4$  
$\displaystyle S$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}D.$  

Solving this simultaneously gives $S=42$ as the age of the son and $D=84$ as the age of Diophantus.


Pappas, T. ``Diophantus' Riddle.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 123 and 232, 1989.

© 1996-9 Eric W. Weisstein