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Diophantus: the Ancient Mathematician and his Riddle

Diophantus greek mathmatician riddle

Diophantes was a Hellenistic Greek mathematician who lived around 200 AD. His claim to fame comes from substituting symbols for numbers and operations in equations, thus creating algebra, but everything else we know about his life comes from a single algebraic riddle.

Here it is:

“Here lies Diophantus.

God gave him his boyhood one-sixth of his life;

One twelfth more as youth while whiskers grew rife;

And then yet one-seventh ‘ere marriage begun.

In five years there came a bouncing new son;

Alas, the dear child of master and sage,

After attaining half the measure of his father’s life, chill fate took him.

After consoling his fate by the science of numbers for four years, he ended his life.”

For those keeping score, that means

x/6 + x/12 + x/7 + 5 + x/2 + 4 = x

Obviously, x equals the number of years Diophantus lived. This equation also assumes that Diophantus’ son died at an age equal to half his father’s ultimateage (transcribed in the equation above, in bold, as “x/2”), as opposed to Diophantus’ age at the time of his son’s death.

Simplifying the equation gives us:

25x/28 + 9 = x

Which we can clean up in a few steps:

25x = (x-9) x 28

25x = 28x – 252

3x = 252

x = 84

And that means Diophantes died at 84.

This “biography”, if you will, was first written down in the Greek Anthology, compiled by Metrodorus around 500 AD. It may not be wholly accurate, but it is fun. And at least we have some of Diophantes’ books of algebra, titledArithmetica, which included innovations like using symbols for commonly-used operations, and substituting symbols for numbers. They also include problems that demonstrate Diophantes’ methods.

A minority of his work survives—only six of 13 complete books—but the fact that some of his problems show up in books written in Arabic show that he was widely-read in his own time and afterwards.