"Here lies Diophantus, the wonder behold..."
Purplemath
In fifth-century Greece, or thereabouts, a grammarian named Metrodorus collected "epigrams", which were generally short poems which had been inscribed somewhere, such as on someone's tomb. He published an anthology of mathematical puzzlers, each in the form of a poem.
One of these poems relates to the life, and the age at death, of a third-century mathematician named Diophantus, who lived in or around Alexandria, Egypt (but was probably of Greek heritage). Diophantus wrote a seminal series of books called the "Arithmetica", and is regarded by many as being "the father of algebra".
Despite all the genuinely new mathematics that Diophantus did (including creating the field of study which would later come to be called "Diophantine equations"), most algebra students know him only from Metrodorus' poem, in various English translations. Students are expected to dissect the puzzler to figure out how long Diophantus lived.
Here's how that works:
- "Here lies Diophantus," the wonder behold...
Through art algebraic, the stone tells how old:
"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, this dear child of master and sage,
Attained only half of his father’s full age.
When chill fate took him — an event full of tears —
Heartbroken, his father lived just four more years."
How long did Diophantus live?
My first task is to translate the poetry from the headstone into practical terms. Anybody solving this puzzler will have to make some guesses and assumptions; here is a set of assumptions that makes the math work out in the end:
- I will assume that "boyhood" stands for pre-adolscent childhood, especially since "whiskers" are mentioned later. So he spent one-sixth of his life in this period.
- "Youth while whiskers grew" could stand for pubescence (that is, the teenage years, into young adulthood); he spent one-twelfth of his life in this period.
- "Ere marriage began" might stand for "unmarried adulthood" or "bachelorhood"; he spent one-seventh of his life in this period.
- There were five years between his wedding and the time his first child was born.
- Tragically, this child died young, living only half as long as his father eventually would; looked at the other way, half of Diophantus' life was spent while his child was alive.
- Diophantus died four years after burying his child.
First things first. I am supposed to figure out Diophantus' age, so I'll pick the variable d stand for Diophantus' age at death. Working with this variable, I can then create expressions for each of the listed periods in Diophantus' life.
childhood:
adolescence:
bachelorhood:
childless marriage: 5
age of child at death:
life after child's death: 4
His whole life had been divided into intervals which, when added together, give the sum of his life. So I'll add the lengths of those periods, set their sum equal to his (as-yet unknown) total age, and solve:
Looking back at the beginning of my work, I am reminded that the variable "d" stands for "how long Diophantus lived", so my complete answer is:
Diophantus lived to be 84 years old.
You can check the answer if you like, by plugging "84" into the original problem. If he lived to be 84, then one-sixth of his life is 14 years, one-twelfth of his life is 7 years (so he'd be 21, and he certainly should have a beard by this age), one-seventh of his life is 12 years (so he didn't marry until he was 33 years old), his child was born when he was 38, the boy died at 42 (when Diophantus was 80), and then Diophantus died four years later.
Always try to label your variables and expressions clearly, as this will go a long way toward helping you get your equations set up correctly. And remember that you can always check your answers (like I did on the last example above); checking your answers is an especially good idea on tests.
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