Last night I had the boys read the book again and write down three things that they found interesting. This morning we talked through some of those ideas. I’d forgotten that my older son had an appointment this morning, so we were unexpectedly pressed for time. So, sorry if parts of this project feel a bit rushed – I think I panicked a bit more about the time than I should have.

Here’s what my younger son had to say about the book – he was interested in a few of the twists and turns that happened in Hilbert’s hotel:

My older son interpreted my instructions in a different way and came up with a few conjectures instead. In this video we talk about whether or not the complex numbers (with only integer coefficients) could fit inside of Hilbert’s hotel.

This was a pretty lucky break as my younger son had wondered about the rationals, which is essentially the same problem:

Finally, we discussed the real numbers and the boys both guessed that they wouldn’t fit in. At the end I showed them Cantor’s diagonal argument . . . just in time for my older son to head out!

Two things:
1) It’s sad that this artist doesn’t have more kids’ books. There’s another one about trolls which is awesome.
2) You’re trying to catch a wayward robot wandering in the plane. At time zero it starts at integer coordinates (x,y) and each second it moves a constant displacement (a,b), also integer. Your process of “catching” it means you get to visit some spot (x(t),y(t)) at each positive time t and see if it’s there, then. Show that there’s a sequence (x(t),y(t)) of places to look that will successfully catch any robot.

Two things:

1) It’s sad that this artist doesn’t have more kids’ books. There’s another one about trolls which is awesome.

2) You’re trying to catch a wayward robot wandering in the plane. At time zero it starts at integer coordinates (x,y) and each second it moves a constant displacement (a,b), also integer. Your process of “catching” it means you get to visit some spot (x(t),y(t)) at each positive time t and see if it’s there, then. Show that there’s a sequence (x(t),y(t)) of places to look that will successfully catch any robot.

Reminds me a little of the famous rabbit and hunter problem from the 2017 IMO:

https://artofproblemsolving.com/wiki/index.php/2017_IMO_Problems/Problem_3

It’s funny how different it is, though!