Inventory007:  Amalgamation Nim (Proposed by Stephen C. Locke, 11/9/01).

In regular Nim, the players take any non-zero number of objects from any one pile. There is a winning strategy based on the base two representations of the sizes of the piles. (Aliens who work in base two might wonder why we think the game has any difficulties.)
In splitting Nim (due to Laskar -- but I have no further reference yet), players are allowed the additional move of splitting a pile into two smaller piles. Again, there is a strategy based on the base two representations of a simple function of the sizes of the piles.
In class last week, Dawne Richards asked about allowing a player to amalgamate two piles in addition to the regular Nim moves. [No splitting allowed in this version -- we don't want cycles in the game graph.] A player may either take from some pile or merge two piles.
What is the winning strategy for amalgamation Nim? (I can write it down for the 3-pile game.)

Discussion. [SCL, 11/9/01]:  In general, it might not be as easy as for the previous two forms I mentioned: in those games, the sum of two losing games is a losing game. I'm don't think this is true in amalgamation Nim. Thus, Conway's Nimbers might not be directly useful. I didn't see the game mentioned in the two Conway books I have access to. That doesn't mean it isn't there, just that I didn't see it.
 
 
 

Bibliography.
 
 
 

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