The following procedure is a variant of one suggested by
Patrek Ragnarsson (age 10). Let $M$ be a finite multiset of
integers. A *move* consists of choosing two elements
$a\neq b$ of $M$ of the same parity and replacing them with the
pair $\frac 12(a+b)$, $\frac 12(a+b)$. If we continue to
make moves whenever possible, the procedure must eventually
terminate since the sum of the squares of the elements will
decrease at each move. What is the least and the most number
of moves to termination, in particular, if $M=\{1,2,\dots,
n\}$? If $M=\{a_1,\dots,a_n\}$, then an upper bound on on
the maximum number of moves is $\frac 12\sum (a_i-k)^2$,
where $k$ is the integer which minimizes this sum. (In fact,
$k$ is the nearest integer to $\frac 1n(a_1+\cdots+a_n)$.)

We can turn this procedure into a game by having Alice and Bob move alternately, with Alice moving first. The last player to move wins. (We could also consider the misère version, where the last player to move loses.) Which multisets are winning for Alice, especially $M=\{1,2,\dots,n\}$? The game is impartial, so it has a Sprague-Grundy number. However, it doesn't seem to be useful for analyzing the game since a position $M$ never breaks up into a disjoint union (or sum) of smaller independent positions. Nevertheless we can ask for the Sprague-Grundy number of a position $M$.

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