World
Game
Of
Sprouts
Association

Genus Sequences

GRUNDY NUMBERS

The normal-play Grundy number of a game G, written Grundy⁺(G) is the unique nonnegative integer x such that o⁺(G+*x) = P. (Again, the superscripted plus signs here designate functions; the nonsuperscripted plus sign here indicates the concatenation of game components.)

Examples:

1. Grundy⁺(*0) = 0
2. Grundy⁺(*1) = 1
3. Grundy⁺(*1+*1) = 0
4. Grundy⁺(S1) = 0
5. Grundy⁺(S2) = 0
6. Grundy⁺(S2+*1) = 1

Similarly, the misere-play Grundy number of a game G, written Grundy^-(G) is the unique nonnegative integer x such that o^-(G+*x) = P.

Examples:

1. Grundy^-(*0) = 1
2. Grundy^-(*1) = 0
3. Grundy^-(*1+*1) = 1
4. Grundy^-(S1) = 1
5. Grundy^-(S2) = 0
6. Grundy^-(S2+*1) = 1

POP QUIZ

Fill in the blanks in the following:

1. Grundy⁺(*2+*2) = ?
2. Grundy^-(*2+*2) = ?
3. Grundy⁺(*2+*3) = ?
4. Grundy^-(*2+*3) = ?

GENUS SEQUENCES

The Genus of a game G, written Genus(G) is the unique infinite sequence of nonnegative integers a,b,c,d,e,... written a^bcde... such that

1. a = Grundy⁺(G)
2. b = Grundy^-(G)
3. c = Grundy^-(G+*2)
4. d = Grundy^-(G+*2+*2)
5. e = Grundy^-(G+*2+*2+*2)
6. etc.

Examples:

1. Genus(*0) = 0^1202020...
2. Genus(*1) = 1^0313131...
3. Genus(0P0) = 0^1202020...
4. Genus(0L0) = 1^0313131…
5. Genus(S2) = 0^0202020...

NIM-VALUES

The first two elements of Genus(G) = a^b are referred to in Winning Ways, by Berlekamp, Conway and Guy, as the "nim-values" of G. For example, the nim-values of S2 are 0⁰.

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