As it turns out, 4 spots is not half-tame, but the pattern does continue: 4+4+4....+4 (with n 4s) seems to be a misere losing position for n >= 3 (verified by Aunt Beast for 1 <= n <= 10). There should be a direct way to prove that this losing trend continues indefinitely, since 4 spots is equivalent to the game {*3,{*1,*2,{*3,{*2}}}}, as I will show below."[4 spots is] not the nimber 1, corresponding to the one-counter heap, but a half-tame game as 4 + 4 game has value 0^0. News : in 2007 it was shown both by Josh Purinton and Danny Purvis that the 4 + 4 + 4 game is a misere losing position. There are strong evidences that this pattern continues for any sum of 4 spots components."
Here's how to show that 4 spots is half-tame:
- 2 spots = {*2} is tame with nim-values 0^0.
- 2P1 = {*3,{*2}} is tame with nim-values 1^1.
- 3L0 = 2L1 = {*1,*2,{*3,{*2}}} is restive with nim-values 0^3, and is
also half-tame since every option is tame, and it has an option with
nim-values 1^0 (*1) and an option with nim-values 1^1 ({*3,{*2}}).
- 4 spots = {*3,{*1,*2,{*3,{*2}}}} is wild (because its tame option *3
doesn't suffice to determine its nim-values) with nim-values 1^0, and
is also half-tame since every option is either tame or half-tame, and
it has no option with nim-values 0^1 or 1^0.
Since 4 spots is half-tame, we can conclude, by the Half-Tame theorem,
that 4 spots + 4 spots is tame of genus 0^0.
On 6/1/10 12:40 PM, Jean-François Peltier wrote:
> ... converges at heap-7 (7 instances of the starting component 4-spots)
Actually, the heap game is calculated by linearizing the game
*[(2[/1]21)3],
and defines only seven sizes of heaps:
heap-1 = *[1] (player may remove one token),
heap-2 = *[2] (player may remove one or two tokens),
heap-3 = *[3] (player may remove one, two, or three tokens),
heap-4 = *[2/] (player may remove two tokens),
heap-5 = *[2[/1]] (player may remove one or two tokens),
heap-6 = *[2[/1]21] (player may remove one, four, or five tokens), and
heap-7 = *[(2[/1]21)3] (player may remove one or four tokens).
So convergence at heap-7 actually means convergence at any number of copies
of heap-7, and is the only kind of convergence possible. That's what
happens when you calculate the misère quotient of a game (or a finite
number of games) rather than of a HeapRule that can apply to arbitrarily
large heaps.
Dan