## Sunday, September 13, 2015

### Riddle Me This, Diophantus

Blog post for Doing Math

Diophantus was a great, ancient greek mathematician, who played a very important role in the subject of algebra (https://en.wikipedia.org/wiki/Diophantus). To honor this man, my MTH 495 professor gave us a riddle - a riddle of algebra - to solve that would give Diophantus' age when he died. Here it is, as given on the class handout, below:

" '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.' "

The language seems clear but riddles are tricks of the mind. What seems straightforward, isn't, and the challenge is finding where the riddle deliberately leads the mind astray, so it would make an erroneous assumption. In other words, how has the riddle tricked us into thinking one interpretation, when it really meant something else?

To demonstrate how challenging the riddle was for me, I went through 7 attempts or 7 different interpretations of the riddle before I got the correct answer. For the first attempt, I tried solving the riddle without using algebra, and that meant picking an age for Diophantus, and doing the series of computations laid out in the riddle to see if it was correct. Basically, guess-n-check. I picked 50 because people rarely lived for very long back then.

On this first attempt, I thought the challenge, or the trick in the riddle, was in the line "And then one-seventh ere marriage begun." The word ere means before, according to merriam-webster.com (http://www.merriam-webster.com/dictionary/ere). So, I had to subtract 1/7 from his boyhood and youth before continuing on with addition. So, the sequence of events laid out in the riddle is not in chronological order. Everything else seemed straightforward. I interpreted "God gave him one-sixth of his life," as 1/6 of his age. "One-twelfth more as youth," I interpreted as his youth was 1/12 more than his boyhood, or 1/6 of his age. "One-seventh ere marriage begun," I took as 1/7 of youth subtracted from his boyhood and youth. My interpretation for the rest: then 5 years after his marriage, a son was born, who grew to half the age that Diophantus eventually reached, 4 years after his son's death. As mentioned before, I picked 50 for Diophantus' age. So, I did the following computations:

Since the ages don't match, my guess was wrong. But the closeness between the two numbers gives a false impression that 50 was close. However, the fractions would divide any number in such a way that would make the two numbers seem close. At this point, I decided that I could keep on blindly guessing or try to solve the riddle in the most efficient way possible, with algebra.

When I continued with my other attempts, I realized the challenge was no longer with the chronological wording of the riddle. The trick lay with the wording of the fractions. In my middle school teaching course, the class went over fractions, and whenever someone said "Take 1/2," our professor would always say "1/2 of what?" The fraction needs to be applied to another number, but sometimes the riddle doesn't tell explicitly what the fraction is being applied to. For example, nowhere in the line "One-twelfth more as youth" does it say what 1/12 is being taken from. Is it 1/12 more of his entire life that is being added to his boyhood, or is it  1/12 of his boyhood, or is it just simply 1/12 of his life was his youth, and not in addition to his boyhood?

I interpreted the riddle differently, with different algebraic equations, 6 more times. On the last attempt, I got it right. I will show the different combinations or ways I interpreted the riddle with the variations for each section of Diophantus' lifespan below. The straightforward interpretations are on top and the multiple interpretations are on the bottom:

Finally on the last, overall 7th attempt, I got the answer right, at least according to Wikipedia (https://en.wikipedia.org/wiki/Diophantus). The interpretation necessary was youth equaling his boyhood plus 1/12 of his entire life, and his marriage was 1/7 of his boyhood before the end of his youth. My solution is below:

I enjoyed this riddle very much. It reminded me how the process of mathematics should be like. Doing math is about looking at problems or situations with different interpretations to see how it plays out and what's discovered in the process. After all, learning requires making mistakes. Figuring out what went wrong is necessary to find out what will make it right.

Figures were created using Microsoft Word and Geogebra.

#### 1 comment:

1. Very thorough think aloud. 5Cs +.

The only think I can think of to add is how you could know in the guess and check whether you were too high or too low. And your comment about the fractions would make any guess seem close - what do you mean?

But definitely nice work!