I found another error in my previous sprouts postings. Let "biosphere" denote a part of a sprouts position sealed off from the remainder of the position. For many years I believed that: (The Mystic Principle) if a given biosphere, played as a separate game, produces n survivors both in normal and misere sprouts, then it must produce n survivors with best play as a component of any game. I reasoned that either player can force n survivors and that n must be good for one or the other. This reasoning is apparently faulty, however.
If a biosphere is played as a game both normally and in misere form, there are 4 possible outcomes. I have an idiosyncratic naming convention for this pattern:
Normal Play Misere
Null Player 2 wins Player 1 wins
Inverter Player 1 wins Player 2 wins
Trap Player 2 wins Player 2 wins
Switch Player 1 wins Player 1 wins
The next chart shows what a particular type of biosphere can become in exactly one move. "1" means "can become"; "0" means "cannot become"; "*" is a wildcard, standing for "1" or "0".
N I T S
Null 0 0 0 0 0 1 0 *
Inverter 1 0 0 *
Trap 0 0 0 1
Switch 1 1 * * 0 1 1 * 1 0 1 * 0 0 1 *
Notice that 16 "varieties" of biopheres are distinguished, including 10 "varieties" of switches. The final 6 varieties of switches listed - i.e. the final 3 lines of the chart - I call "pretraps".
Sprague-Grundy numbers are integers that can be used to determine which biosphere needs to be altered. A biosphere which if played as a normal game is lost for the first player has a Sprague-Grundy number of 0. A biosphere which in exactly one move can become a biosphere with Sprague-Grundy number m, such that m is any value from 0 to n but not n+1, has Sprague-Grundy number n+1.
My faulty "Mystic Principle" amounted to an assertion that a null must have a Sprague-Grundy number of 0 (which is true, by the way) and that an inverter must have a Sprague-Grundy number of 1 (which is false). So, let me apologize and correct any previous misstatements with the following chart. I'll also promise not to post any more sprouts-related things unless I run across more mistakes that I have already posted. (Of course, the crash of my "Mystic Principle" might impinge on some of the move by move analysis I posted previously.)