You’ll want to read the first post to catch up.

So, I’ve decided to call these scenarios “balanced non-transitive dice”. I’ve yet to prove that balanced non-transitive n-sided dice exist for n>2. It should be straightforward, but a simple construction would be nice. At any rate, I’m convinced of this.

The issue beyond is the following: Most solutions so far have a victorious probability of: (floor((n^2)/2)+1)/(n^2). This is fancy language for “as close to half as possible”. The first post mentions this. I have no doubt that this is a sharp bound for n=3. This means that a set of balanced non-transitive 3-sided dice exist with victorious probability 5/9 (it’s written in the first post), but that a set does not exist with victorious probability 6/9.

A professor of mine ran a program for n=4, with interesting results. The only balanced sets he found had victorious probability 9/16. Unless his code has an error, the bound is sharp for n=4.

I then did some diagram chasing by hand, and found, for n=5:

A: 15, 9, 8, 7, 1

B: 14, 13, 6, 5, 2

C: 12, 11, 10, 4, 3

A beats B beats C beats A 14/25 of the time. The bound is broken! Then, just now:

A: 15, 14, 5, 4, 2

B: 13, 12, 11, 3, 1

C: 10, 9, 8, 7, 6

A beats B beats C beats A 15/25 of the time! Now this is cool.

More to come.