Tanya Khovanova shared some work with a bunch of high schoolers (whoa!) on patterns in Cookie Monster games. Cookie Monster games are Nim games where k sticks (cookies) can be removed from multiple heaps. Each ruleset has a different restriction on which sets of piles can be removed from.
She and her students considered rulesets where you can take from...:
- No restriction
- One-or-all piles
- One-or-two piles
- Any consecutive piles (assuming the piles are in a list)
- One or two consecutive piles
- Any set of piles including the first jar
- Any odd number of piles. (It turns out that the P positions are the same as in Nim!)
- Any set of piles except all of them.
- ... and more!
In one of these games, she noticed a surprising correlation with an automaton problem! A sequence of the number of P positions is related to the number of new cells born from the Ulam-Warburton automaton. She then looked more closely at other relationships to automaton!