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Topic: The Game of Sprouts
Replies: 6   Last Post: Jan 19, 1998 5:49 PM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: the game of Sprouts
Posted: Jan 19, 1998 5:49 PM
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On 19 Jan 1998, Danny Purvis wrote:

> zovich - the nim building block; null - a zovich which will yield an
> even number of moves; inverter - a zovich which will yield an odd
> number of moves; switch - a zovich which the mover can convert to her
> choice of null or inverter; trap - a zovich which will become a switch
> after one move; biosphere - a hermetically sealed Sprouts life zone;
> countable - a Sprouts dot with 0 or 2 lines attached.
> A Sprouts zovich with an even (odd) number of countables which will
> yield an odd (even) number of biospheres is an inverter. An inverter
> plus a trap wins.
> These concepts are very useful because of a ubiquitous trap, the
> biosphere consisting of two unattached dots.

You've discovered one-infinieth of the Sprague-Grundy theory!
This applies when games decomose as sums of smaller ones (for
Sprouts these are your "biospheres", and assigns to each summand
a number called it's "nim-value" or "Sprague-Grundy number". The
definition is that a position has nim-value n just if you can move from it
to positions of all nim-values from 0 to n-1, but not to n. A game that
necessarily lasts for a fixed number of moves has nim-value
0 or 1 according as this number is even or odd (your "null"s
and "inverter"s).

However, there are other positions that have nim-values 0 and 1,
as well as those that have the larger nim-values 2,3,4,... . You
can find some tables of nim-values in Sprouts in "Winning Ways"

John Conway

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