Advanced Authentic Research

Analyzing the Combinatorial Game the Princess and the Roses by Sarah Y.

Summary

The project is centered around the combinatorial game The Princess and the Roses. In this game, there are n heaps which each have some chips in them. On a move, a player may either remove one chip from a heap or two chips from different heaps. There are two players, and the last person to make a move wins. The game has been solved for up to 5 heaps and some cases of the 6 heap problem has been solved. The aim of this project is to continue this analysis using a function that generates a number for any given position of a game, the Grundy number, which can identify which player would win from a given position. Grundy numbers can be calculated recursively, so in this project a computer program will be made that recursively calculates the Grundy number for larger and larger positions for a given number of heaps. The project aims to verify the results previously found for up to 5 heaps and find new results for 6 or more heaps empirically from the program and to form conjectures based on these patterns. These conjectures will then be proved mathematically. With time permitting, generalizations of the game will also be studied.