With Just 2 Squares

Probability Level 3

What is the maximum number of regions that 2 unit squares can divide a plane into?

In the above images, we see that the 2 unit squares divide the plane into 4 regions: one overlapping region, two non-overlapping regions,and the region outside the two squares.

4 6 8 10

1 solution

Chung Kevin
Aug 14, 2016

We can get 10 regions as above.

I don't know how to prove that this is the maximum, though I'm convinced that it is.

It would appear that, in general, two identical regular $n$ -sided polygons can divide the plane into a maximum of $2(n + 1)$ regions.

It gets much trickier if we were to add another square to this problem. My first attempt yielded $26$ regions, but I'm far from sure of that being the answer.

Brian Charlesworth - 4 years, 10 months ago

I believe the first statement is true :)

Chung Kevin - 4 years, 9 months ago

With Just 2 Squares

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