View source for Kayles

{{Short description|Mathematical game}}
{{About|the combinatorial game|the lawn game|Bowling|and|Skittles (sport)}}
[[File:Bowling ball and pins for strike - front view.jpg|thumb|A row of bowling pins. On their turn, a player may choose to eliminate a single pin, or two adjacent ones.]]

'''Kayles''' is a simple [[impartial game]] in [[combinatorial game theory]], invented by [[Henry Dudeney]] in 1908. Given a row of imagined bowling pins, players take turns to knock out either one pin, or two adjacent pins, until all the pins are gone. Using the notation of [[octal games]], Kayles is denoted '''0.77'''.

== Rules ==
Kayles is played with a row of tokens, which represent bowling pins. The row may be of any length. The two players alternate; each player, on his or her turn, may remove either any one pin (a ball bowled directly at that pin), or two adjacent pins (a ball bowled to strike both). Under the [[normal play convention]], a player loses when they have no legal move (that is, when all the pins are gone). The game can also be played using [[misère]] rules; in this case, the player who cannot move ''wins''.

== History ==
Kayles was invented by [[Henry Dudeney]].<ref>{{citation|first=H. E. |last=Dudeney|title=The Canterbury puzzles|isbn=0-486-42558-4|pages=118–119, puzzle 73|publisher=Dover|year=2002}}. Originally published in 1908.</ref><ref name="onag">Conway, John H. ''On Numbers and Games.'' Academic Press, 1976.</ref> [[Richard K. Guy|Richard Guy]] and [[Cedric Smith (statistician)|Cedric Smith]] were first to completely analyze the normal-play version, using [[Sprague–Grundy theorem|Sprague-Grundy theory]].<ref name="guy-smith">R. K. Guy and C. A. B. Smith, The ''G''-values of various games, Proc. Cambridge Philos. Soc., 52 (1956) 514–526.</ref><ref name="plambeck">T.E. Plambeck, [http://www.plambeck.org/oldhtml/mathematics/games/misere/MISEREPA.PDF Daisies, Kayles and the Sibert-Conway decomposition in misere octal games] {{Webarchive|url=https://web.archive.org/web/20100714132904/http://www.plambeck.org/oldhtml/mathematics/games/misere/MISEREPA.PDF |date=2010-07-14 }}, Theoret. Comput. Sci (Math Games) (1992) 96 361–388.</ref> The [[misère]] version was analyzed by [[William Sibert]] in 1973, but he did not publish his work until 1989.<ref name="plambeck2">{{citation |author=Plambeck, Thane |title=Kayles |url=http://www.plambeck.org/oldhtml/mathematics/games/misere/077/index.htm |access-date=2008-08-15 |archive-url=https://web.archive.org/web/20081012123544/http://www.plambeck.org/oldhtml/mathematics/games/misere/077/index.htm |archive-date=2008-10-12 |url-status=dead }}</ref>

The name "Kayles" is an Anglicization of the French ''[[wikt:quilles|quilles]]'', meaning "bowling pins".

== Analysis ==
Most players quickly discover that the first player has a guaranteed win in normal Kayles whenever the row length is greater than zero. This win can be achieved using a [[symmetry strategy]]. On their first move, the first player should move so that the row is broken into two sections of equal length. This restricts all future moves to one section or the other. Now, the first player merely imitates the second player's moves in the opposite row.

It is more interesting to ask what the [[nim-value]] is of a row of length <math>n</math>. This is often denoted <math>K_n</math>; it is a [[nimber]], not a [[number]]. By the [[Sprague–Grundy theorem]], <math>K_n</math> is the [[mex (mathematics)|mex]] over all possible moves of the [[nim-sum]] of the [[nim-value]]s of the two resulting sections. For example,

: <math>K_5 = \mbox{mex}\{K_0 + K_4, K_1 + K_3, K_2 + K_2, K_0 + K_3, K_1 + K_2\},\,</math>

because from a row of length 5, one can move to the positions

: <math>K_0 + K_4,\quad K_1 + K_3,\quad K_2 + K_2,\quad K_0 + K_3,\text{ and } K_1 + K_2.\,</math>

Recursive calculation of values (starting with <math>K_0 = 0</math>) gives the results summarized in the following table. To find the value of <math>K_n</math> on the table, write <math>n</math> as <math>12a + b</math>, and look at row a, column b:

{| class="wikitable" style="text-align:center; margin-left: auto; margin-right: auto; border: none;"
|+ Kayles nim-values through <math>K_{83}</math>
|-
!<math>K_n</math>
! width=20|0
! width=20|1
! width=20|2
! width=20|3
! width=20|4
! width=20|5
! width=20|6
! width=20|7
! width=20|8
! width=20|9
! width=20|10
! width=20|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
|}
At this point, the nim-value sequence becomes periodic<ref name="plambeck2" /> with period 12, so all further rows of the table are identical to the last row.

== Applications ==

Because certain positions in [[dots and boxes]] reduce to Kayles positions,<ref name="ww">[[E. Berlekamp]], [[J. H. Conway]], R. Guy. ''[[Winning Ways for your Mathematical Plays]].'' Academic Press, 1982.</ref> it is helpful to understand Kayles in order to analyze a generic dots and boxes position.

==Computational complexity==

Under normal play, Kayles can be solved [[Polynomial time|in polynomial time]] using Sprague-Grundy theory.<ref name="guy-smith" />

==Generalizations==

''Node Kayles'' is a generalization of Kayles to [[Graph (discrete mathematics)|graphs]] in which each bowl “knocks down” (removes) a desired vertex and all its neighboring vertices. Alternatively, this game can be viewed as two players finding an [[Independent set (graph theory)|independent set]] together. Winner determination is solvable in polynomial time for any family of graphs with bounded asteroidal number (defined as the size of a largest subset of vertices such that the removal of the closed neighborhood of any vertex in the set leaves the remaining vertices of the set in the same connected component).<ref>{{cite journal|last1=Bodlaender|first1=Hans L.|title=Exact Algorithms for Kayles|journal=Theoretical Computer Science|date=2015|volume=562|issue=|pages=165–176|doi=10.1016/j.tcs.2014.09.042|doi-access=}}</ref>

Similarly, in the ''clique-forming'' game, two players must find a [[Clique (graph theory)|clique]] in the graph. The last to play wins. Schaefer (1978)<ref>{{cite journal|last1=Schaefer|first1=Thomas J.|title=On the complexity of some two-person perfect-information games|journal=Journal of Computer and System Sciences|date=1978|volume=16|issue=2|pages=185–225|doi=10.1016/0022-0000(78)90045-4|doi-access=}}</ref> proved that deciding the outcome of these games is [[PSPACE-complete]]. The same result holds for a partisan version of node Kayles, in which, for every node, only one of the players is allowed to choose that particular node as the knock down target.

== See also ==
* [[Combinatorial game theory]]
* [[Octal games]]
* [[Octal games#Dawson's Kayles|Dawson's Kayles]]
* [[Nimber]]

== References ==
<references />

[[Category:Combinatorial game theory]]
[[Category:Mathematical games]]