World
Game
Of
Sprouts
Association

Two New (Old) Misere Theorems

I have guarded these two theorems for over one year!

Theorem 1

Gx vs Gx vs 6k+4 spots = G0

Gx vs Gx vs 6k+7 spots = G0

Where

1. 1 < x < 4
2. Gx+ = Gx- ("+" normal game,"-" misere game)

Example:

PP2 vs PP2 vs 10 spots = G0
Gx = PP2 = G3
Gx+ = PP2+ = Gx- = PP2-

Theorem 2

Gx vs Gy vs 6k spots = G0

Where

1. x, y > 1
2. x = y
3. Gx+ = Gx- ("+" normal game),"-" misere game)

Example:

PP2 vs 1L2 vs 6 spots = G0
Gx = PP2 = G3
Gy = 1L2 = G3
Gx+ = PP2+ = Gx- = PP2- (Gx = PP2 normal game = PP2 misere game = G3)

Notes:

"vs" indicates the juxtaposition of biospheres.

"G" refers to Grundy number, either normal or misere as indicated.

Please see notation section of glossary for other abbreviations.

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