Disjunction and existence properties: Difference between revisions

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[[Image:Hexapawn.png|200px|right|thumb|A 3×3 hexapawn board]]
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'''Hexapawn''' is a [[deterministic game|deterministic]] two-player [[game]] invented by [[Martin Gardner]]. It is played on a rectangular board of variable size, for example on a 3×3 board or on a [[chessboard]]. On a board of size ''n''×''m'', each player begins with ''m'' [[pawn (chess)|pawn]]s, one for each [[Square (geometry)|square]] in the row closest to them. The goal of each player is to advance one of their pawns to the opposite end of the board or to prevent the other player from moving.
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Hexapawn on the 3&times;3 board is a [[solved game]]; if both players play well, the first player to move will always lose. Indeed, Gardner specifically constructed it as a game with a small [[game tree]], in order to demonstrate how it could be played by a [[heuristic]] [[Artificial intelligence|AI]] implemented by a [[mechanical calculating device|mechanical computer]]. A variant of this game is [[octapawn]].
  <li>[http://www.imc-e.net/thread-1293613-1-1.html http://www.imc-e.net/thread-1293613-1-1.html]</li>
 
 
==Rules==
   <li>[http://bbs.shenghuoqu.com.cn/forum.php?mod=viewthread&tid=1275410&fromuid=83620 http://bbs.shenghuoqu.com.cn/forum.php?mod=viewthread&tid=1275410&fromuid=83620]</li>
As in [[chess]], each pawn may be moved in two different ways: it may be moved one square forward, or it may capture a pawn one square diagonally ahead of it.  A pawn may not be moved forward if there is a pawn in the next square. Unlike chess, the first move of a pawn may not advance it by two spaces. A player loses if he/she has no legal moves or the other player reaches the end of the board with a pawn.
 
 
  <li>[http://www.bv58.cn/news/html/?10462.html http://www.bv58.cn/news/html/?10462.html]</li>
==Dawson's chess==
 
Whenever a player advances a pawn to the penultimate rank (unless it is an [[isolated pawn]]) there is a threat to proceed to the final rank by capture. The opponent's only sensible responses are therefore either to capture the advanced pawn or to advance the threatened one, the latter only being sensible in the case that there is one threatened pawn rather than two. If one restricts 3&times;<math>N</math> hexapawn with the additional rule that the capture is always compulsory, the result is the game '''Dawson's chess'''.
  <li>[http://playmond.com/activity/p/46783/ http://playmond.com/activity/p/46783/]</li>
 
 
Dawson's chess reduces to the [[impartial game]] denoted '''.137''' in Conway's notation. This means that it is equivalent to a [[Nim]]-like game in which:
  <li>[http://www.sociland.com/index.php/blogs/22589/191035/abercrombie-australia-outlook http://www.sociland.com/index.php/blogs/22589/191035/abercrombie-australia-outlook]</li>
*on a turn, the player may remove one to three objects from a heap,
 
*removing just one object is a legal move only if the removed object is the only object in the heap, and
</ul>
*when removing three objects from a heap of five or more, the player may also split the remainder into two heaps.
The initial position is a single heap of size <math>N</math>.
The [[nim-sequence]] for this game is
 
0.1120311033224'''0'''5'''22'''3301130211045'''2'''74'''0'''
   1120311033224455'''2'''33011302110453748
  1120311033224455933011302110453748
  1120311033224455933011302110453748
  1120311033224455933011302110453748 ...,
 
where bold entries indicate the values that differ from the eventual periodic behavior of the sequence.
 
==References==
* Mathematical Games, ''[[Scientific American]]'', March 1962, reprinted in ''The Unexpected Hanging and Other Mathematical Diversions'', by Martin Gardner, pp. 93ff
* ''[[Winning Ways for your Mathematical Plays]]''
 
==External links==
* [http://www.chessvariants.org/small.dir/hexapawn.html Hexapawn] - an article by Robert Price.
* [http://www.javazoid.com/hexapawn.html Hexapawn java applet] - source code included.
 
[[Category:Mathematical games]]
[[Category:Chess variants]]
[[Category:1962 in chess]]
[[Category:Board games introduced in 1962]]

Latest revision as of 15:59, 17 May 2014

Nike Store Australia are absolutely adorable

Sores can show up all over the body but mostly around the mouth and nose. Impetigo can be spread through close contact or by sharing such things as towels and toys. It includes 20low income senior units on 4 above grade levels plus an 1,800 sf commercial component on the floor level. There are 16 subterranean and one grade Nike Store Australia level parkingspaces in the project.

Winmner of Warrnambool to Melbourne Rosd RIvtc, League Cop, and sores of Other Iaces. The Mascotte bicycle, known throughout Australias " "The Good Luck Wheel," bas proved its merit time upon time. But a couple of things are going on here. Firstly, Dell can be quite cheap.

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The first thing that you will notice if you played World of Warcraft before is that all the talent trees have become smaller. In the original World of Warcraft, talent trees had 7 tiers culminating in an ultimate ability at the bottom. Some sites approach this by providing users almost full treatments for the way their. It seems like every day there is a new technology being added which can make the internet more accessible.

The numbers don't add up. An 82 percent increase in the chief branch's operating budget. Our community one of the best resources for all iPhone News, iPhone Questions, iPhone Help, software Reviews and general iPhone Discussions. The first step would be to REGISTER so you can take advantage of all the features we have to offer.

And, most important of all, GET SOME SLEEP. Sleep deprivation is among the worst causes of depression.. Lowballing the price allows the investment banks to reward big funds. Those funds often are big customers of the banks, using Pandora Australia them to help trade stock, design custommade derivatives bets and provide other services..

But the incremental price of an online transaction is basically zero. It does not cost anything to execute a line or two of code once the code's been written. There are villagers, who mysteriously disappear during the telling, and sibling hiphop minions from the evil queen, who wear funky baseball hats and pants pulled down to the floor, who dance gloriously but have nothing whatsoever to do with the story. The animals, who peek out of the foliage, are absolutely adorable: the brightplumed tropic bird includes a tail of straw; Nike Free Run the giraffe rises high off to the side of the stage; and the elephant, though surprisingly small in stature, includes Michael Kors Melbourne a proboscis with a life of its own..

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