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. TheWinning Waysargument

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.

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