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One Can Never Have Too Many Problems

I am always on the lookout for good, interesting problems involving mathematics. The recent issue of the newsletter L'Augarithms, edited by Professor Kenneth Kaminsky at Augsburg College, included a problem that caught my eye.

Begin with coins arranged in piles. At each turn, rearrange the coins according to the following rule: remove the top coin from each pile, possibly eliminating piles, and form a new collected pile of coins. The game continues until you revisit a previously encountered (unordered) arrangement, having reached a terminal cycle.

What are the fixed points, if any?
Where are two cycles, if any?
Which states are cyclic?
And what happens when we generalize to two players?

Note: This problem formed the basis of a talk given by Professor Su Doree as part of a colloguium. The title of the talk: The Coins Go 'Round 'n 'Round: Bulgarian Solitaire and Exchange.


Source: L'Augarithms, April 21, 2010

Hint: Sorry...I am still exploring this problem myself. I believe it best to build some small piles of coins, make the rule-based moves, look for patterns, and work from there to larger piles.


Solution Commentary: If you need resolution (while I continue to explore), you can find some answers by doing a Google search on "Bulgarian Solitaire." For example, the web site entry on Wikipedia summarizes relevant responses to some of the questions raised, while the website MathCentral provides better insights and connects the game to a magic trick. Another resource is Ethan Akin and Morton Davis's article "Bulgarian Solitaire". American Mathematical Monthly (1985) 92(4): 237250.