# Fox and Geese Museum

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).

### Summary of Results

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.
• No position has atomic weight greater than 5.  The only values of atomic weight 5 are `1^5*` (at height 8) and `2^5*` (at heights 9 and 11).
• Only four values have negative atomic weights: `2v*`, `5/2v`, `4v`, and `4vv*`.
• The only values with non-integer atomic weights are `{2^3*|2*}` (atomic weight `3/2`), `{2^4|2}` (atomic weight `2*`), and `{2^5*|2*}` (atomic weight `{3|2}`).
• The only negative right stops that appear are off and -1.  -1 first appears at height 20.
• The only values of form `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.
• The only loopy value that appears, aside from various combinations of over and off, is the game `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`.
• The "most complicated" value that appears, judged by the rather crude standard of "longest output string", is the following position of "length" 216:
`{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`
• There are only eight non-loopy pure infinitesimals.  They are summarized in the following table, indexed by the height at which they first appear.
 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*|*||*`
• There are a great many loopy pure infinitesimals that show up.  They were not included in the official results, since (a) they are of less intrinsic interest, and (b) they are so numerous that their inclusion would have more than doubled the size of the output file!  Several are quite complicated, such as:
`{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