en>Dawnseeker2000: Reverted edits by 1.38.25.173 (talk) to last version by Dawnseeker2000

2013-06-02T16:49:52Z

Reverted edits by 1.38.25.173 (talk) to last version by Dawnseeker2000

← Older revision		Revision as of 18:49, 2 June 2013
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	~~My name~~ is ~~Christy Belcher but everybody calls me Christy~~. I'~~m from France~~. I'~~m studying~~ at the ~~college~~ (~~final year~~) ~~and I play~~ the ~~Clarinet for 3 years~~. ~~Usually I choose music from my famous films~~ ;). <br>~~I have two brothers~~. ~~I love Amateur geology~~, ~~watching TV~~ (~~Grey~~'~~s Anatomy~~) and ~~Trainspotting~~.<br><br>~~Also visit my weblog~~ - [http://~~ielaw~~.~~uibe~~.~~edu.cn~~/~~wiki~~/index.~~php~~/~~Top_10_Marketing_Strategies_for_Real_Estate_Investors_Part_5_Classified_Ads marketing advertising promotions~~]		In [[combinatorial game theory]], '''Kayles''' is a simple [[impartial game]]. In 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 he or she has 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''.

			== 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 his or her 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:

			<!-- Can anyone make this table look nice? -->
			<center>
			{\| class="wikitable" style="text-align:center"
			\|+ 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
			\|}</center>
			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 ==
			<!-- Expand this! -->
			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.

			== 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 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], 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">Plambeck, Thane. "Kayles." http://www.plambeck.org/oldhtml/mathematics/games/misere/077/index.htm</ref>

			The name "Kayles" is an Anglicization of the French ''quilles'', meaning "bowling."

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

			== References ==
			<references />

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

204.244.11.74 at 22:42, 16 March 2012

2012-03-16T22:42:14Z

New page

My name is Christy Belcher but everybody calls me Christy. I'm from France. I'm studying at the college (final year) and I play the Clarinet for 3 years. Usually I choose music from my famous films ;). <br>I have two brothers. I love Amateur geology, watching TV (Grey's Anatomy) and Trainspotting.<br><br>Also visit my weblog - [http://ielaw.uibe.edu.cn/wiki/index.php/Top_10_Marketing_Strategies_for_Real_Estate_Investors_Part_5_Classified_Ads marketing advertising promotions]

Power reverse dual currency note - Revision history

en>Dawnseeker2000: Reverted edits by 1.38.25.173 (talk) to last version by Dawnseeker2000

204.244.11.74 at 22:42, 16 March 2012