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how many different rings can you make given the constraints?

There are at least three, maybe more.

The rules:

you may assume the pattern repeats all the way around the circle.

You may alter the number of segments, but there must be an even number of segments so that there are an equal number of squares and trapezoids

C may equal either A or B, but does not have to equal either.

The ONLY dimensions you can use are 24, 33, 36, 42, 45, 48, 54, 60, 72, 84, 96, 108, 120

For easy mode, you can make the the inner ring be all the same. (i.e. b,b,b,b in stead of a,b,a,b,a,b) in this case there may be an odd number of segments)

For easy mode #2 you can eliminate B all together and just have all C

merge at a single point in the center.

**bonus question**

if A=60, B=54and C=36, does it work? if so, how many segments are there? How big is the circumfrence?