How many ways can circles overlap

This activity was inspired by this numberphile video by Neil Sloane.

Here are the only distinct three ways you can draw two circles in a Venn diagram.

If we take the universal set to be the positive integers, we can come up with some sets that would fit for A and B for the three examples.

1. A = Odd numbers, B = Multiples of 4
2. A = Even numbers C= Prime numbers
3. A= Even numbers B=Multiples of 4

Note that these choices mean that there is at least one number in each region.

Now for the challenge …

Draw all the different Venn diagrams you can make with three circles, and find rules for them all.

Number of ways can be found here. A picture of them all is here.

Watch the numberphile video to find out how many Jonathan Wild found for 4 and 5 circles. It is a lot more! Nobody knows for 6 …