By Blair Seidler
This project is Chapter 4 of my dissertation. I plan to split it out into a separate PDF which will be uploaded here when it is done.
In a beautiful 2022 paper (in French), Julien Lemoine solved the problem of which values of n admit a winning strategy for player one. For this project, I explored the Sprague-Grundy values of the game for the basic game and several variations.
The "Easy" version allows the first player to choose any number. This version is not interesting from a winning strategy persepctive because the first player can select a prime greater than n/2 for most values of n. It is at least mildly interesting from a Sprague-Grundy perspective, however.
As in the basic game, we have a board numbered with the integers from 1 to n. In this variation, we have two additional game parameters, positive integers a and b. Player one starts the game by selecting an even number on the board. Players then alternate selecting previously unused numbers which are factors or multiples of the current number, with the additional restriction that the largest permissible divisor is a and the largest permissible multiplier is b.
As in the basic game, we have a board numbered with the integers from 1 to n. In this variation, we have two additional game parameters, sets of positive integers A and B. Player 1 starts the game by selecting any number on the board. We do not restrict the initial selection to an even number. Divisibility is relevant in the basic game, but it is not relevant in the additive scenario. Players then alternate selecting previously unused numbers by subtracting an element of A from the current number c or adding an element of B to c.