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| The '''octal games''' form a significant subclass of [[impartial game]]s studied in [[combinatorial game theory]].<ref name="WW">{{cite book
| | If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. Keep reading for some great Word - Press ideas you can start using today. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br> |
| | last=Berlekamp
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| | first=Elwyn R.
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| | authorlink=Elwyn Berlekamp
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| | coauthors=John H. Conway, Richard K. Guy
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| | title=[[Winning Ways for your Mathematical Plays]]
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| | publisher=Academic Press
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| | year=1982
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| | isbn = 0-12-091101-9, ISBN 0-12-091102-7}} Revised and reprinted as<br />
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| {{cite book
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| | author=---
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| | title=[[Winning Ways for your Mathematical Plays]] (2nd ed.)
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| | publisher=A K Peters Ltd
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| | date=2001--2004
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| | isbn = 1-56881-130-6, ISBN 1-56881-142-X, ISBN 1-56881-143-8, ISBN 1-56881-144-6}}</ref><ref>{{cite book
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| | last = Conway
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| | first = John Horton
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| | authorlink = John Horton Conway
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| | title = [[On numbers and games]]
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| | publisher = Academic Press
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| | year = 1976
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| | isbn = 0-12-186350-6}} Revised and reprinted as<br />
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| {{cite book
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| | last = ---
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| | authorlink = John Horton Conway
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| | title = [[On numbers and games]]
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| | publisher = A K Peters Ltd
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| | year = 2000
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| | isbn = 1-56881-127-6}}
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| </ref> They are organized by a numeric coding system that enables a compact specification of a variety of game rules. | |
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| ==Octal games specification==
| | Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. Here is more regarding [http://miniURL.fouiner.info/backup_plugin_740643 wordpress backup] take a look at our own web-site. Some of the Wordpress development services offered by us are:. It sorts the results of a search according to category, tags and comments. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. <br><br>But before choosing any one of these, let's compare between the two. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them. <br><br>Numerous bloggers are utilizing Word - Press and with good reason. But the Joomla was created as the CMS over years of hard work. Websites that do rank highly, do so becaue they use keyword-heavy post titles. There are many advantages of hiring Wordpress developers for Wordpress project development:. The popularity of Word - Press has increased the demand for Word - Press themes and these themes sells like hot cake on the internet. <br><br>Website security has become a major concern among individuals all over the world. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. ) Remote Login: With the process of PSD to Wordpress conversion comes the advantage of flexibility. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
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| An octal game is played with tokens divided into heaps. Two players take turns moving until no moves are possible. Every move consists of selecting just one of the heaps, and either
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| * removing all of the tokens in the heap, leaving no heap,
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| * removing some but not all of the tokens, leaving one smaller heap, or
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| * removing some of the tokens and dividing the remaining tokens into two nonempty heaps.
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| Heaps other than the selected heap remain unchanged. The last player to move wins in ''normal play''. The game may also be played in [[misère game|misère play]]'', in which the last player to move loses.
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| Games played with heaps in this fashion, in which the allowed moves for each heap are determined by the original heap's size, are called ''Taking and Breaking games'' in the literature.<ref name="WW"/> Octal games are a subset of the taking and breaking games in which the allowed moves are determined by the number of tokens '''removed''' from the heap.
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| The octal code for a game is specified as
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| :0 . ''d''<sub>1</sub> ''d''<sub>2</sub> ''d''<sub>3</sub> ''d''<sub>4</sub> …, | |
| where the octal digit ''d''<sub>''n''</sub> specifies whether the player is allowed to leave zero, one, or two heaps after removing ''n'' tokens from a heap. The digit ''d''<sub>''n''</sub> is the sum of
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| * 1 if leaving zero heaps is permitted, 0 otherwise;
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| * 2 if leaving one heap is permitted, 0 otherwise; and
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| * 4 if leaving two heaps is permitted, 0 otherwise.
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| Zero tokens are not counted as a heap. Thus the digit ''d''<sub>''n''</sub> is odd if a heap of ''n'' tokens may be removed entirely, and even otherwise. The specification of one-heap results in ''d''<sub>''n''</sub> applies to removing ''n'' tokens from a heap of more than ''n''. The two-heap results in ''d''<sub>''n''</sub> apply to removing ''n'' tokens from a heap of at least ''n''+2, and separating the remainder into two nonempty heaps.
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| Octal games may allow splitting a heap into two parts without removing any tokens, by use of the digit 4 to the left of the decimal point. This is similar to the move in [[Grundy's game]], which is to split a heap into two unequal parts. Standard octal game notation, however, does not have the power to express the constraint of unequal parts.
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| Octal games with only a finite number of non-zero digits are called ''finite octal games''.
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| ==Particular octal games==
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| ===Nim===
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| The most fundamental game in [[combinatorial game theory]] is [[Nim]], in which any number of tokens may be removed from a heap, leaving zero or one heaps behind. The octal code for Nim is '''0.333…''', appearing in the published literature as
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| :<math>0.\dot{3}</math> ,
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| to signify the repeating part as in a [[repeating decimal]]. It is important to realize, however, that the repeating part does not play the same role as in octal fractions, in that the games | |
| :<math>0.0\dot{7}</math>
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| and
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| :<math>0.1</math>
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| are not identical, despite their equality as octal fractions.
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| ===Kayles===
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| The game [[Kayles]] is usually visualized as played with a row of ''n'' pins, but may be modeled by a heap of ''n'' counters. One is allowed to remove one or two tokens from a heap and arrange the remainder into zero, one, or two heaps. The octal code for Kayles is '''0.77''' .
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| ===Dawson's Chess===
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| ''Dawson's Chess'' is a game arising from a chess puzzle posed by [[Thomas Rayner Dawson]] in ''Caissa's Wild Roses'', 1938.<ref>{{cite book
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| | last=Dawson
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| | first=Thomas Rayner
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| | authorlink=Thomas Rayner Dawson
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| | title=in Five Classics of Fairy Chess
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| | publisher=Dover
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| Publications
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| | year=1973}}</ref> The puzzle was posed as involving opposed rows of pawns separated by a single rank. Although the puzzle is not posed as an [[impartial game]], the assumption that captures are mandatory implies that a player's moving in any file results only in the removal of that file and its neighbors (if any) from further consideration, with the opposite player to move. Modeling this as a heap of ''n'' tokens, a player may remove an entire heap of one, two, or three tokens, may reduce any heap by two or three tokens, or may split a heap into two parts after removing three tokens. Dawson's Chess is thus represented by the octal code '''0.137'''.
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| == Nim-sequence ==
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| The [[Sprague–Grundy theorem]] implies that a heap of size n is equivalent to a [[Nim|nim heap]] of a given size, usually noted G(n). The analysis of an octal game then consists in finding the sequence of the nim-values for heaps of increasing size. This sequence G(0), G(1), G(2) ... is usually called the nim-sequence of the game.
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| All ''finite'' octal games analyzed so far have shown a nim-sequence ultimately periodic, and whether all finite octal games are ultimately periodic is an open question. It is listed by [[Richard Guy]] as an important problem in the field of [[combinatorial game theory|combinatorial games]].<ref>Richard K. Guy, Unsolved problems in Combinatorial Games, Games of No Chance, 1996</ref>
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| == Computation records ==
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| A complete analysis of an octal game results in finding its period and preperiod of its nim-sequence. It is shown in [[Winning Ways for your Mathematical Plays]] that only a finite number of values of the nim-sequence is needed to prove that a finite octal game is periodic, which opened the door to computations with computers.
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| Octal games with at most 3 octal-digits have been analyzed through the years. There are 79 non-trivial octal games, among which 14 have been solved :
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| * .156 by Jack Kenyon in 1967<ref name="WW"/>
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| * .356, .055, .644 and .165 by Richard Austin in 1976<ref name="WW"/>
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| * .16, .56, .127 and .376 by Anil Gangolli and Thane Plambeck in 1989<ref name="WW"/>
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| * .454, .104, .106, .054 and .354 by Achim Flammenkamp between 2000 and 2002<ref name="AF">Achim Flammenkamp, [http://wwwhomes.uni-bielefeld.de/achim/octal.html Octal games]</ref>
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| There remain 63 of these games, despite the computation of millions of nim-values by Achim Flammenkamp.<ref name="AF"/>
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| ==See also==
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| Octal games like [[Nim]], in which every move transforms a heap into zero or one heaps, are called [[Quaternary numeral system|quaternary games]] because the only digits that appear are 0, 1, 2, and 3. The octal notation may also be extended to include [[Hexadecimal numeral system|hexadecimal games]], in which digits permit division of a heap into three parts. In fact, arbitrarily large bases are possible. The analysis of quaternary, octal, and hexadecimal games show that these classes of games are markedly different from each other,<ref name="WW"/> and the behavior of larger bases has not received as much scrutiny.
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| Some octal games with different codes are closely related to each other. In the game '''0.07''', called ''Dawson's Kayles'', a move is to remove exactly two tokens from a heap and to distribute the remainder into zero, one, or two heaps. Dawson's Kayles is named for its (non-obvious) similarity to Dawson's Chess, in that Dawson's Kayles heap of ''n''+1 tokens acts exactly like a Dawson's Chess heap of ''n'' tokens. Dawson's Kayles is said to be a '''first cousin''' of Dawson's Chess.
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Octal Game}}
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| [[Category:Combinatorial game theory]]
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| [[Category:Mathematical games]]
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