Difference between revisions of "Draw"
From HexWiki
(some corrections and link for Brouwer's Fixed Point Theorem) |
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* A [http://www.cs.ualberta.ca/~javhar/hex/hex-galeproof.html proof by David Gale] that used the fact that exactly three hexes meet at every vertex. | * A [http://www.cs.ualberta.ca/~javhar/hex/hex-galeproof.html proof by David Gale] that used the fact that exactly three hexes meet at every vertex. | ||
| − | * | + | * An [http://www.cs.ualberta.ca/~javhar/hex/hex-yproof.html elegant proof] using the [[Y|game of Y]]. |
| + | * Another [[Y#No draws|proof]] using the game of Y. | ||
In fact, the no-draw property is equivalent to the 2-dimensional case of [http://en.wikipedia.org/wiki/Brouwer_fixed_point_theorem Brouwer's fixed point theorem] (a non-trivial theorem from topology saying that any continuous map from the unit square into itself must have a fixed point). | In fact, the no-draw property is equivalent to the 2-dimensional case of [http://en.wikipedia.org/wiki/Brouwer_fixed_point_theorem Brouwer's fixed point theorem] (a non-trivial theorem from topology saying that any continuous map from the unit square into itself must have a fixed point). | ||
Revision as of 17:25, 9 November 2007
One of the beautiful properties of Hex is that the game can never end in a draw, i.e., there is always a winner.
There are various ways of proving this, for example:
- A proof by David Gale that used the fact that exactly three hexes meet at every vertex.
- An elegant proof using the game of Y.
- Another proof using the game of Y.
In fact, the no-draw property is equivalent to the 2-dimensional case of Brouwer's fixed point theorem (a non-trivial theorem from topology saying that any continuous map from the unit square into itself must have a fixed point).
