The discovery of the solution to misere Kayles

September 21, 2003 at 4:54 pm (kayles, papers, solutions)

Conway describes normal play Kayles on pg 127 of On Numbers and Games, Academic Press, 1976:


Kayles is the octal game 0.77. It was first completely solved in normal play by Richard Guy, and by Sibert and Conway for misère play.

Normal Play Nim Sequence

                    0  1  2  3  4  5  6  7  8  9 10 11
                    ----------------------------------
                0+  0  1  2  3  1  4  3  2  1  4  2  6
               12+  4  1  2  7  1  4  3  2  1  4  6  7
               24+  4  1  2  8  5  4  7  2  1  8  6  7
               36+  4  1  2  3  1  4  7  2  1  8  2  7
               48+  4  1  2  8  1  4  7  2  1  4  2  7
               60+  4  1  2  8  1  4  7  2  1  8  6  7
               72+  4  1  2  8  1  4  7  2  1  8  2  7

The Kayles nim sequence is periodic of length 12—the values in final row of the table repeat themselves indefinitely.

Sibert-Conway Decomposition for Misère Play

The surprising solution to the misère version of Kayles was discovered by the amateur William L. Sibert in 1973, but it was not published until over seventeen years later. There’s an interesting story behind these events. In 1989, Sibert wrote a description of his solution and a proof in an unpublished 43 page document entitled The Game of Misere Kayles: The “Safe Number” vs “Unsafe Number” Theory. Sibert wrote the following “Preamble” to this document:

Some years ago (about 1961) I was stumped by a problem in an old puzzle book (possibly by Dudeney). It described the plight of a group of tourists in the Alps who were consistently defeated by a young Swiss miss at a pluck-the-petals-from-the-daisy type of game.

Two players took turns plucking petals, and the game required each player to take either one or to petals at a time … with the proviso that if two petals were taken, they had to physically adjacent.

The winner was the player who took the last petal, and, according to the author, the young lass always won … whether she played first or second.

The “solution” given in the back of the book described her strategy as one of presenting her opponent with a daisy which had been divided into two identical segments. She then simply matched her opponent’s play each time, usng the sector opposite the one into which the opponent had just played.

This struck me as an unsatisfactory answer, in that she had to rely on inept play by her opponent in those cases where she played first.

This led me, for some reason, to try and find the winning strategy for a re-defined game in which the wind had randomly blown away a number of petals from a large daisy before the game began.

(It was only much later that I learned that the problem I had set for myself was to find the solution to the well-known [normal play] game “Kayles”.)

After many, many, many hours of work, I had the strategy for all possible games in which the largest unbroken string of petals was 168 or less… and was satisfied that this strategy could be applied to any game with a string or strings exceeding 168. (I could have stopped at 166, but did the next two numbers just to round out a final cycle of 12).

The work was done on a commuter train, returning home from work in the evenings, and I used worksheets which bore a crude resemblance to the Grundy scale described in “Winning Ways” … except that my worksheets didn’t “slide”. A copy of one of those worksheets is attached, as Appendix IV.

Having solved the problem, I forgot about it until I happened to see a Martin Gardner column in an issue of Scientific American in which he discussed Kayles. The issue came out in 1969 or 1970, and, as I recall, his number values matched mine exactly, except for one number (28 ?). I rechecked by calculations, and concluded that the variance was almost certainly caused by a “typo” in the article. Many years later, when I acquired a copy of “Winning Ways” my values were confirmed as correct.

In any case, reading the Gardner article reawakened my interest in Kayles, and I set out to try and solve the problem of the Misere version of the game.

In time, I developed the theory of “Safe” and “Unsafe” numbers … and by 1973 I had what I believed was a general solution to the game.

Again, I set the matter aside, but for some reason in 1979 I sent an outline of my solution to Mr. Gardner, asking him whether it was correct. He replied that he didn’t know, and suggested that I check with Professor Guy.

Once more I let matters slide, but in 1989 I finally sent the “solution” to Professor Guy, and asked for his reaction. The subsequent exchange led me to assemble my work papers into what I trust is a coherent document. What follows is the document wich I hope will confirm, under scrutiny, that my theory is correct.

Here is the solution as presented in the paper by Sibert and Conway, Mathematical Kayles:

The PN Positions of Kayles (ie those that are P-positions in normal play, and N-positions in misere play) are precisely those positions that have one of the following three forms:


E(5) E(4,1)
E(17,12,9) E(20,4,1)
25 E(17,12,9) D(20,4,1)

While the N-normal, P-misere positions are the

NP Positions
D(5) D(4,1)
E(5) D(4,1)
E(4,1) D(9)
12 E(4,1)
E(17,12,9) D(20,4,1)
25 D(9) D(4,1)

The notation E(a, b, . . .) (resp. D(a, b, . . .)) refers to any position composed by taking an even (resp., odd) number of isolated rows of pins of length of size a or b or . . ..

For example,

5 + 4 + 1 + 1
is a position included in the set

D(5) D(4, 1)
since it is composed by taking a single (ie, odd number) row of size 5 and the total number of 4’s or 1’s (ie, three) is also odd. For every other position in misere Kayles not listed in the forms of PN- and NP-positions above, its misere and normal play outcomes agree.

It’s possible to frame this solution using the language of misere quotients. The misere quotient of Kayles is a commutative monoid of order 40, and its misere pretending function is also of period 12, just as its normal play nim sequence has period twelve. This paper gives the details.

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