**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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