Download The Chooser-Picker 7-in-a-row game

The Chooser-Picker
7-in-a-row game
András Csernenszky
July 3 2008
Szeged
Hypergraph games
(an example)
•The vertices of the graphs are the fields of an
infinite graph paper.
•The winning sets are the horizontal, vertical or
diagonal consecutive cells of length 5.
•If one of the player (because of the strategy
stealing argument [J.Nash], this is the first
player) gets a length 5 line, then he wins
otherwise the game is draw.
Given an F = (V, F) hypergraph and two players: the first
player and the second player. The players alternate each
other choosing one of the hypergraph’s edge. We call
these games as hypergraph games.
Hypergraph games
 If both player plays well then who wins
that game?
 Is it possible that no one wins?
 What happens if we change the length of
the winning sets?
Changing the rules:
the „Weak” Games
(Maker–Breaker games)
If Breaker
wins the
„weak”occupies
game, then
the original
 •Theorem:
Maker (~first
player) wins:
if he/she
a winning
set game
draw. (~second player) wins: if he/she prevents Maker’s win.
 is
Breaker
hasonly
weaker
chance
to winoutcomes
than the second
in the original game.
 •~Breaker
-There are
two
possible
of theplayer
game.
Original game
The first player (x)
need to block the 2nd
player offence.
„Weak” game
Twofold threat!
-Maker (x) wins
Changing the rules: the
Chooser-Picker games
 Beck’s
Picker conjecture:
picks two vertices,
and wins
Chooser
one of them, the other
If Breaker
the chooses
Maker-Breaker
one remains
to Picker.
game,
then also
Picker (as a second player) wins the Chooser Picker
Chooser
wins by getting a full winning set, and Picker wins if he prevents
game.
this (as the in Maker-Braker games).
~These games are close to each other. If we believe that Breaker
 If there is odd number of vertexes, then the last one goes to Chooser by
wins
the game, then the Chooser-Picker version can be analyzed
definition.
(and also Picker win expected)
„Weak” game:
Maker wins!
Chooser-Picker game
Chooser wins!
Using Chooser-Picker games
(an example: 4x4 Tic-Tac-Toe)
•We think that this game is draw.
•If we could prove that Breaker wins the
weak version of this game then we are
ready.
•We can check quickly whether the
Chooser-Picker version of this game is
a Picker win?
•And now we should start the more time-consuming
proof; that Breaker also wins that game…
•The size of the game tree is the same, but if we
know a winning strategy for Picker we can prove is
somewhat easier.
About the k-in-a-row
games



The first player wins for k = 5 on the 19×19
or even in the 15 × 15 board [Allis]
The first player wins if k<=4, and the game
is a blocking draw if k >=9 [Shannon and
Pollak]
k=8 is also a draw [T. G. L. Zetters]
OPEN questions:
•k=5 on infinite board?
•k=6, 7?
k=7
 The original game is believed to be a
draw.
 It would be stronger to prove that the
weak version is a Breaker win.
 At first we examine the Chooser-Picker
version of this game.
The Chooser-Picker 7-in-a-row-game:
An auxiliary game
 We consider a tiling of the plane, and play an
auxiliary-game on each tile (sub-hypergraph).
 It is easy to see, if Picker wins all of these subgames, then Picker wins the game played on any K
board which is the union of disjoint tiles.
The Chooser-Picker 7-in-a-row-game:
Tricks-1
 If Picker wins the Chooser-Picker game on (V,F), then Picker also
wins it on (V \ X,F(X))
The Chooser-Picker 7-in-a-row-game:
Tricks-2
 If in the course of the game (or just already at the beginning) there
is a two element winning set {x, y} then Picker has an optimal
strategy starting with {x, y}
The Chooser-Picker 7-in-a-row-game:
Tricks-3
 It helps Chooser’s game if we change one of Picker’s square to a
free square, and it is also advantageous for Chooser if he/she gets
one of the free squares (P « FREE«C).
Playing on the 4x8 board…
Results:
 Picker wins the Chooser-Picker 7-in-a-row (case study
on the 4x8 auxiliary board, few pages).
 It is an additional information for the original 7-in-a-row
game.
 We checked M-B game on the same auxiliary board,
but it is a Maker win! (brute force computer search)
 We should find other auxiliary games (and check the
C-P version at first).
Thank you for your
attention!