Genus theory classification of the Lemoine-Viennot list of reduced games with up to 6 initial spots

[Josh Jordan]

Here is a classification by Genus theory of the Lemoine-Viennot list of reduced games corresponding to every sprouts position reachable from an initial position of 6 spots. I am open to other suggestions, but for now I will refer to their dataset as "LV6". Judging by the rate of growth in the number of reduced games as the number of initial spots increases (see [1], page 15), it will be a while before anyone creates the corresponding dataset for 7 spots.

In what follows, I have used the terminology of CGSuite <http://cgsuite.sourceforge.net/docs/api/org/cgsuite/impartial/Phylum.html>. Note that a single reduced game may correspond to many sprouts positions, but each sprouts position corresponds to only one reduced game.

Of the 25458 games in LV6 (*):

  14895 are Wild of Restless Genus,

   2910 are Wild of Tame Genus,

   2373 are Wild,

   2246 are Reversibly Tame,

   2157 are Wild of Restive Genus,

    564 are Reversibly Restive,

    128 are Hereditarily Tame,

    115 are Restless,

     61 are Hereditarily Restive,

      8 are nim heaps (*0 through *7), and

      1 is Tameable.

Of the initial positions with 0-6 spots,

  0 spots is *0,

  1 spot is *0,

  2 spots is Hereditarily Tame with nim-values 0^0,

  3 spots is *1,

  4 spots is Wild of Restive Genus with nim-values 1^0,

  5 spots is Reversibly Tame with nim-values 1^1, and

  6 spots is Tameable with nim-values 0^1.

Of the games in LV6:

   3990 have a normal-play Grundy number of 3,

   3904 have a normal-play Grundy number of 5,

   3660 have a normal-play Grundy number of 4,

   3658 have a normal-play Grundy number of 2,

   2456 have a normal-play Grundy number of 1,

   2292 have a normal-play Grundy number of 0,

   2221 have a normal-play Grundy number of 7,

   2193 have a normal-play Grundy number of 6,

    827 have a normal-play Grundy number of 8,

    231 have a normal-play Grundy number of 9,

     25 have a normal-play Grundy number of 10, and

      1 has a normal-play Grundy number of  11.   

The lone game with a normal-play Grundy number of 11 is Wild of Restless Genus with nim-values 11^1. It corresponds to the position [0.1<<1<<<2>>>>>].

Of the games in LV6:

   8510 have a misere-play Grundy number of  0,

   8466 have a misere-play Grundy number of 1,

   2680 have a misere-play Grundy number of 2,

   2499 have a misere-play Grundy number of 3,

   1213 have a misere-play Grundy number of 4,

   1172 have a misere-play Grundy number of 5,

    457 have a misere-play Grundy number of 7,

    440 have a misere-play Grundy number of 6,

     19 have a misere-play Grundy number of 8, and

      2 have a misere-play Grundy number of 9.

The two different reduced games with a misere-play Grundy number of 9 are Wild with nim-values 6^9. One corresponds to [0.AB.CD;2E<AB>G;CD;EG] and the other corresponds to [1<1>.2.B;1CBD;CD].

(*) Though the paper says there are 25459 games in the list, I see only 25458.

[1] Lemoine & Viennot. Analysis of misere Sprouts game with reduced canonical trees (2009).  http://arxiv.org/abs/0908.4407

[Josh Jordan]

Oops, when I wrote,

  4 spots is Wild of Restive Genus with nim-values 1^0,

I should have written:

  4 spots is Wild of Tame Genus with nim-values 1^0.

I invite others to check my work. LV6 is available at <http://sprouts.tuxfamily.org/wiki/doku.php?id=misere>.

The files I used are:

- n6posrct-393103.spr which gives, for each sprouts position, the id of its corresponding game, and

- n6rctch-25482.spr which gives, for each game id, the ids of its options.