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Thanks to Gary Antonick of Numberplay for inviting me to contribute once more, with a problem on cutting a pizza that is somewhat related to my previous post on dividing the plane.

Rather than give any spoilers for that problem, I’ll offer a few other related problems.

Venn diagrams are traditionally drawn with circles.  With 0 circles, you have 1 region.  With 1 circle, 2 regions.  With 2 circles, 4 regions.  Why do we stop with 3 circles, 8 regions?

I’ve seen some interesting Venn diagrams for 5 sets. Most of them use ellipses:

Five ellipses make a Venn diagram

while others use triangles, while some use …polyominoes?  Let’s see, 5 sets will need 32 regions – is that the most we can make using ellipses?  Using triangles?  Why do I see these diagrams with 5 sets and not with 6 or more?  You can make Venn diagrams with more than 5 sets, but they look pretty weird! Although when colored properly, these Venn diagrams can be quite beautiful.

The next post in this series will answer the questions about how many regions can be created using these various shapes and pose some new ones.  Several of these images are from a great article about symmetric Venn diagrams — I’m more concerned here about counting the number of possible regions, like in the Numberplay column, rather than in ensuring that we have a Venn diagram (where every possible intersection of sets exists exactly once).

Symmetry is often thought of as something that applies to geometry, but of course it can be found in algebra or even probability.

The 2007 AMC8 gave us a deck of cards with a red A, B, C, D and a green A, B, C, D, and winning hands of two cards were two of the same color or two of the same letter.

What is the probability of a winning hand?

We can do this problem the hard way: there are 8 choose 2 = 28 hands, there are 4 choose 2 = 6 red pairs, 4 choose 2 = 6 green pairs, and 4 same-letter pairs, so that makes 16/28 = 4/7.

But we can also use symmetry.  Your first card can be anything.  Whatever it is, there are 3 cards of the same color and 1 of the same letter out of the remaining 7 cards, so you have a 4/7 chance of winning on your second pick.  Because the given cards and the rules for winning don’t distinguish between the two colors or the four letters, every first card has this same chance of winning.  That’s symmetry!


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