## There’s convincing and then there’s curiosity

April 1st, 2012 — ddelmoliThis past week, Cary Millsap posted one of the ways he’s teaching his children to understand concepts and to prove to him that they understand the methods for solving problems by presenting them with a classic puzzle.

The puzzle talks about the length of a piece of string wrapped around a circle (in this case, the whole Earth), and how much longer it would need to be if it was raised 4 more inches off the ground? As his kids worked through the problem, Cary provided us with the neat answer in which you didn’t need to know the circumference of the Earth, or any other circle for that matter — the increase in length is a simple function applied to the additional height above the ground.

Near the end of Cary’s post, he talks about how the understanding of basic relationships and methods gives the ability to convince someone that you actually know what you are talking about and goes on to challenge you to demand the same information from people trying to sell you something.

It’s a wonderful post, and I encourage you to go read it — but I also want you to go beyond the original post a bit.

While Cary’s formula holds for circles, what about other kinds of shapes? What about simple regular polygons (where all sides are the same length)? Do they have the same kind of simple formula? Do you have the curiosity to try and see if you can determine methods for generalizing what you know, applying it, and demonstrating to others?

Convincing me makes me trust you with a particular problem, showing me you won’t stop there makes me trust you even further…

I spent the afternoon playing with the idea and I think I’ve found it — I came up with the following answers:

- Equilateral triangle (3 sides) = 41.5692 inches
- Square (4 sides) = 32 inches
- Pentagon (5 sides) = 29.0617 inches
- Hexagon (6 sides) = 27.7128 inches
- Octagon (8 sides) = 26.5097 inches

Looks like it’s tending toward’s Cary’s answer for a circle — perhaps there is a formula after all? :-)

April 6th, 2012 at 2:01 am

[...] to Dominic Delmolino, there’s convincing and then there’s [...]

April 6th, 2012 at 11:12 am

[...] Delmolino wrote a follow-up to Cary’s post, ending with the question about generalising the formula for a circular [...]

April 6th, 2012 at 1:40 pm

Dominic,

Nice follow-up to Cary’s post, which prompted me to get out a pen and an envelope and start doodling. Your last sentence also prompted me into providing an explanation for why the result tends to Cary’s circle formula. The write-up is in an update to: http://jonathanlewis.wordpress.com/2012/03/30/proof-2/

April 7th, 2012 at 3:55 pm

Thanks, Jonathan! Nice work!

February 13th, 2013 at 1:09 pm

[...] to Dominic Delmolino, there’s convincing and then there’s [...]