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 +
1/on

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 + 1/on”

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 Siegel.