# An infinite descent problem with pentagons

The boys had a nice hiking trip to Mt. Osceola today.

When they got back we did a little follow up to yesterday’s project on infinite descent:

Infinite Descent with kids inspired by Jim Propp

That project came from reading the lastest Jim Propp blog post:

Fermatโs Last Theorem: The Curious Incident of the Boasting Frenchman

So, the kids were a little tired tonight, but we still managed to have nice (and short) project tonight discussing this problem from Propp’s blog:

“If you liked the proof that there is no integer-sided golden rectangle, you might try to prove, by similar means, that there is no regular pentagon whose five equal sides and five equal diagonals all have integer length.”

The Zometool set makes this project especially fun!

However, because I am also tired, tonight’s write up will be short (and not edited – I’m going to bed).

In part 1 we built some pentagons and tried to see if we could see a pattern in the lengths of the sides and the diagonals. I’d left the length of the sides of the larges pentagon as a mystery to help motivate finding the pattern:

The boys didn’t see the pattern so we moved to the whiteboard – when you write the lengths down, the patter is much easier to see:

So, next we moved back to the couch to test if the pattern we found worked – it did! My older son wanted say that the longs were an integer and that idea threw me off in our discussion here. To try to get things back on a more correct path (meaning, one that was more clear to me given that I was tired ๐ ), we went back to the witeboard after this part of the discussion:

ok – so back to the white board to see if the boys could tackle the infinite descent proof. They had the idea for how to start the proof in the last video – assume that we’ve found a regular pentagon with integer sides and diagonals. How do we find the smaller pentagons now?

It took a few minutes, but we got there ๐

It is fun seeing kids struggle through a puzzle like this one, and the Zome set is almost a miracle helper for this particular problem. Fun little post hiking project ๐