View source for Small boards

Playing [[Hex]] on [[board]]s of size smaller than 10 &times; 10 is not very interesting, since many players will be able to play almost perfectly. However it may still be interesting for theoretical studies, and for making [[Puzzles|problems]].

The boards of size up to five can be solved by hand. Hex on 6 &times; 6 has been solved by [[Queenbee]]. The board sizes 7 to 9 have been solved with computer programs, too.

Here are the winning first moves on the small boards. [[Red (player)|Red]] is vertical and plays first. The [[Hex (board element)|cells]] shaded red are winning moves for red, while those shaded blue are losing.

== Winner depending on the first move ==
The following boards can help you decide where you should [[swap]] when playing on small boards, and it might give you ideas of patterns for bigger boards.

<hexboard size="2x2"
  coords="show"
  contents="S red:all blue:(a1 b2)"
  />

<hexboard size="3x3"
  coords="show"
  contents="S red:all blue:(a1 b1 b3 c3)"
  />

<hexboard size="4x4"
  coords="show"
  contents="S blue:all red:(a4 b3 c2 d1)"
  />

<hexboard size="5x5"
  coords="show"
  contents="S red:all blue:(a1--d1 a2 a3 b5--e5 e4 e3)"
  />

<hexboard size="6x6"
  coords="show"
  contents="S red:all blue:(a1--e1 a2 b6--f6 f5)"
  />

=== Size 7 ===

Size 7 was first solved by [[Ryan Hayward]] using [[dominated cell|domination]]. The proof tree can be found at http://www.cs.ualberta.ca/~hayward/hex7trees/

<hexboard size="7x7"
  coords="show"
  contents="S red:all blue:(a1--f1 a2 b2 a3 d2 a5 b7--g7 g6 f6 g5 d6 g3)"
  />

=== Size 8 ===

The outcomes for size 8 were computer generated by [[Javerberg]]. The solution was independently computer generated by Hayward et al. and appeared in [[INJCAI|IJCAI09]].

<hexboard size="8x8"
  coords="show"
  contents="S red:all blue:(a1--g1 a2--f2 a3 a4 a6 b8--h8 c7--h7 h6 h5 h3)"
  />

=== Size 9 ===

The outcomes for size 9 by Jakub Pawlewicz and Ryan Hayward.

<hexboard size="9x9"
  coords="show"
  contents="S red:all blue:(a1--h1 d2--g2 a3 a7 b9--i9 c8--f8 i7 i3)"
  />

== Reference ==
* This [http://www.ru.is/faculty/yngvi/pdf/HaywardBJKPR05.pdf article] by Ryan Hayward ''et al.'' is a reference for 7x7.
* This [[Little Golem]]'s forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=338 thread] is a reference for size 8x8.
* This [https://webdocs.cs.ualberta.ca/~hayward/papers/pawlhayw.pdf article] by Jakub Pawlewicz and Ryan Hayward is a reference for size 9x9.

== See also ==

* [[Board size]]
* Yang, Liao, and Pawlak designed a [https://webdocs.cs.ualberta.ca/~hayward/papers/yang7.pdf decomposition method] to find a winning strategy for Hex on small boards.
* For corresponding information on the game of Y, please visit [[Where to swap (y)]].

[[Category: Theory]]