Another T1 “Counterexample”
September 10, 2007
Roman reports that he has found a second example in which the principle which he calls “Theorem 1” or simply “T1” requires the proviso “n>1”. (Aunt Beast found the first example and evidently has confirmed this second example.)
Here is T1:
Gz = (x^y + (3n - 1)spots)- = (x^y + (3n - 1)spots)+
And here is Roman’s new example:
(0L9 Vs. 2 spots)+ =G0
(0L9 Vs. 2 spots)- = Gx x >1
T1 says that a position consisting of two biospheres, where one of those biospheres is comprised of 3n - 1 original spots, where n > 1, is a win for the same player in normal and misere play. (“G” stands for Grundy number, either normal or misere depending upon context.) Roman’s new example shows that T1 would not work without the “where n > 1” proviso. This example is a position of two biospheres in which one biosphere consists of nine original spots inside a loop with two pivots and no spots outside of the loop (the 0L9 biosphere) plus a biosphere comprised of two original spots.
One way to think about this example is to consider 0L9 a “quasiswitch,” a biosphere with preordained parity but which can be forced to contain or not contain a hidden trap as the mover might desire. In this case the parity of the quasiswitch would not be to Left’s total advantage, but in misere sprouts he would still have the advantageous option of moving to the other biosphere. This “trap crashing” option apparently sets up a switch/quasiswitch balance which ultimately works in Left’s favor. (Now if Right adjusts the switch to winning parity for Right, then Left adjusts the quasiswitch to not contain a hidden trap, so that Right loses in misere play. If Right instead adjusts the switch to losing parity for Right, then Left adjusts the quasiswitch to contain the hidden trap, causing Right again to lose in misere (and even in normal) play.) The line perhaps gets complicated with the quasiswitch next evolving to a switch.
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