The Fox and Geese Museum details the results of a search for certain "interesting" values in Fox and Geese. Some of the highlights of this search are listed below, along with a link to the complete museum. The search was limited to positions that fit on a 14x8 board, so superlatives such as "only" and "all" are implicitly qualified with this restriction. However, given the apparent regularity of Fox and Geese values on large boards, it seems unlikely that new "interesting" values will suddenly start to appear on larger boards.
To simplify the presentation we sometimes speak of the "atomic weight" of a
non-infinitesimal game; this just means the atomic weight of its infinitesimal
part (thus, for example, 4vv*
has "atomic weight" -2).
Here are some of the most interesting findings of the museum search:
{0||0|off}
, the tiniest game of all, is quite common, and appears first at
height 8.*
appears first at height 15 and thereafter becomes quite common.^
appears first at height 18.1^5*
(at height 8) and 2^5*
(at heights 9 and 11).2v*
, 5/2v
, 4v
, and 4vv*
.{2^3*|2*}
(atomic
weight 3/2
), {2^4|2}
(atomic weight 2*
), and {2^5*|2*}
(atomic weight
{3|2}
).n + Tiny(G)
are 4Tiny(2*)
, 2Tiny(2)
,
2Tiny(1/2*)
, and 1Tiny(1*)
. Only a single value has form n +
Miny(G)
: 5/2Miny(1/2*)
, which appears exclusively at height 17.G = {3*|G}
. Readers familiar with the
theory of loopy games in Winning Ways will recognize this as 3 - upon
, and
therefore strictly between 3v
and 3vv
. A precise formulation: The game
graph of every Fox and Geese position can be represented as a tree, with every
node equal to one of the following: off
, n + over
for some n, or
3 - upon
.{5/2||||5/2Miny(1/2*)|||2^3*||2|off},{5/2|{3|11/4||5/2,{3*|5/2}|off|||5/2Miny(1/2*)|off||||2over}}|||{5/2Miny(1/2*)|||2^3*||2|off},{3|11/4||5/2,{3*|5/2}|off|||5/2Miny(1/2*)|off||||2over}||1,{1|0}|off||||1,{1|0}|off
15 | * |
18 | ^ |
22 | 0,{1*|0}|0 |
22 | 0||0,{1*|0}|0 |
23 | 0|||0||0,{1*|0}|0 |
24 | 1*|{0|||0||0,{1*|0}|0}||0 |
27 | ^|0,{0|-1} |
28 | 1*|*||* |
{3/2|{2*|3/2||5/4|||1,{1|off}||||1/2}||0,{3/2*|0}|{0|off},{*|off}|||{0|off},{*|off}}
The complete museum is available for download, but be aware that it is quite large!
Aaron Siegel Last modified: 14 May 2003 08:08:46 PM |