# Debra Hydorn

Professor of Mathematics
University of Mary Washington
Fredericksburg, Virginia, USA

I enjoy exploring relationships and patterns using Geometers Sketchpad and R. Most recently I have discovered some interesting patterns associated with the game of Bulgarian Solitaire. The game produces sequences of partitions of an integer n. The elements of the partition are ordered largest to smallest and the next partition is found by subtracting 1 from each element, creating a new element from those subtracted, and reordering largest to smallest. For example, 43211 would be followed by 5321. All such sequences will end in a cycle, regardless of the initial partition. Starting with a partition that is part of a cycle and overlaying that cycle with that of another integer has revealed some regular and irregular patterns.

Partition Cycles for Integers Near 10
18 x 14 cm
Digital Print
2018

This plot shows three separate overlaid plots for the cycle of partitions for two integers around 10, which produces a 1-cycle. The center plot shows the overlaid cycle plots for n=9 (a 4-cycle) and n=11 (a 5-cycle) which has a regular pattern of geometric shapes. The plot above is for n=8 (a 4-cycle) and n=11 and the one at the bottom is for n=9 and n=12 (a 5-cycle). Changing one of the integers changes the plot to one with an irregular pattern, but the series plot for the shared integer with the center plot can be revealed by a careful comparison between the plots.