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Fox & Geese

The game of Fox and Geese is usually played on a cross-like
board. I learned about it from the second volume of the first edition of
Winning Ways
for your Mathematical Plays
which is now reprinted as number 3 of
the series. In the first edition, Elwyn Berlekamp,
John Conway and
Richard Guy claimed that the value of their
starting position (they play it on an 8×8 chess board with the Geese on
places a1,c1,e1 and g1 and the Fox at place e8) has exact value

1 +

where on is the class of all ordinal numbers so
1/on is by far the smallest infinitesimal number you can think
of. In this second edition which I bought a week ago, they write about
this :

We remained steadfast in that belief until we heard
objections from John Tromp. We then also received correspondence
from Jonathan Weldon, who seemed to prove to somewhat higher standards
of rigor that
“The value of Fox-and-Geese is 2 +

Oops! But of course they try to talk themselves out
of it

Who was right? As often happens when good folks
disagree, the answer is “both!” because it turns out that the parties
are thinking of different things. The Winning Ways argument
supposed an indefinitely long board, while Welton more reasonably
considered the standard 8×8 checkerboard.

Anyway, let us be
happy that the matter is settled now and even more because they add an
enormous amount of new material on the game to this second edition (in
chapter 20; btw. if after yesterday you are still interested in the game of sprouts you might be interested in
chapter 17 of the same volume). Most of the calculations were done with
the combinatorial game suite program of Aaron

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