Phelps’s post got me thinking about the expected number of empty squares. So I started playing around and remembered (about an hour later!) that we had done a project for kids on that exact topic before based on this John Allen Paulos tweet:

So, today we revisited that old project. We started by writing a computer program to pick 64 random integers from 1 to 64 and I had the boys plays snap cubes on a chess board on the positions given by the numbers.

Here’s what the boys had to say about this particular random placement of snap cubes:

Next we returned to the code and talked through it. We then looked at what the maximum number of snap cubes on a square was in a few (about 10) different runs.

Now we looked at the number of squares with zero cubes and the number with 1. My older son guessed that the number of squares with zero cubes would be roughly equal to 1/3 of the squares.

Finally, overnight I’d run a program to find the expected number of squares covered. Well, technically something slightly different -> the number of squares that has a 50% probability of being covered.

I then compared that number to (1 – 1/e) times the number of squares and found that the two numbers were nearly the same. That means the number of empty squares is roughly 1/e times the number of squares!

Definitely a fun project for kids. It is a great “hands on” way – and a non-calculus way – to see the number e.