Wednesday, May 18, 2016

Doing Math: The Riddle of Diaphantus

In class we discussed a riddle and I wanted to discuss the way I thought about and solved the riddle.

The riddle: '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, 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.'

Mathematician Diophantus

The only foreseeable strategy I could see was to use algebra. I started off by assuming that "x" was going to equal the length of Diophantus' life. Then I broke the riddle down:

boyhood was 1/6 of life = (1/6)x
grew whiskers was 1/12 of life = (1/12)x
marriage was 1/7 of life = (1/7)x
after 5 years he had a son = 5
child reached half of father's age = (1/2)x
he died 4 years later = 4

Then I added all of these values together and set them equal to x which is the length of Dionphantus' life. Thus we get:

(1/6)x + (1/7)x + (1/12)x +(1/2)x + 5 + 4 = x
(25/28)x + 9 = x
9 = (3/28)x
x = 84

So I believe that Diophantus lived to be 84 years old.

A question posed was: can you solve this problem without using algebra. After discussing with my classmates we came to the conclusion that no it would not be possible. The only way to possibly get an answer would be to assume a length of childhood and extrapolate form there. But even then you will not be able to get a correct answer when using algebra.

An example would be to assume that boyhood was 12 years. Based on that you can assume that life was 72 years long. But now lets add up the values.

boyhood was 1/6 of life = 12
grew whiskers was 1/12 of life = 6
marriage was 1/7 of life = 10.28
after 5 years he had a son = 5
child reached half of father's age = 36
he died 4 years later = 4

This all adds up to 73.28 which is not equal to 72. Therefore the only way to solve this riddle is to

1 comment:

  1. I'm not sure you're finished?

    It seems like to me that the guess of 72 is a good start. Where does it go wrong? How can you tell if the next guess should be more or less? (Some would call this solving with a heuristic.) Which pieces of life are different kind of clues
    than the majority. What do they tell you that's different?

    clear, coherent: +
    content: see note
    consolidated: add a summary or synthesis to tie it together.
    complete: needs more, which is there to add.