IBM Research | Ponder This | September 2020 Solution

September 2020 - Solution

September 2020 Challenge

An RPS(9) game satisfying the requirements must have exactly 81 automorphisms. Here is one example:

 0 -> 1, 2, 3, 5
 1 -> 3, 4, 7, 8
 2 -> 1, 4, 7, 8
 3 -> 2, 4, 7, 8
 4 -> 0, 5, 6, 7
 5 -> 1, 2, 3, 6
 6 -> 0, 1, 2, 3
 7 -> 0, 5, 6, 8
 8 -> 0, 4, 5, 6

An RPS(11) game satsifying the requirements must have exactly 55 automorphisms. Here is one example:

 0 -> 1, 2, 3, 4, 5
 1 -> 2, 5, 7, 9, 10
 2 -> 3, 4, 8, 9, 10
 3 -> 1, 4, 6, 7, 9
 4 -> 1, 5, 6, 8, 10
 5 -> 2, 3, 6, 7, 8
 6 -> 0, 1, 2, 8, 9
 7 -> 0, 2, 4, 6, 10
 8 -> 0, 1, 3, 7, 10
 9 -> 0, 4, 5, 7, 8
 10 -> 0, 3, 5, 6, 9

The games can be analyzed by hand and using group-theoretic arguments, but the solutions can also be found by going over various RPS games and computing the number of automorphisms. A nice discussion of RPS games and their automorphism groups can be found in "Rock-Paper-Scissors Meets Borromean Rings" by Marc Chamberland and Eugene A. Herman.