World

Game

Of

Sprouts

Association

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