## Hex Opening Theory

The Zermelo-von Neumann Theorem

The Nash-Hein Ramsey Theorem

The Topological Lemma (both cannot win)

Anatole Beck

- Tree Games
- Beck's Hex : Does every opening win?
- ``This wrecks Beck's Hex'' : The opening the acute corner loses.

The Central Opening Conjecture (Hayward? Jack van Rijswijck? Berger?)

- Question 1: On an odd sized board, does opening with the center win?
- Question 2: Does each opening on the short diagonal win?

## Hex and Topology

The Nash-Gale Theorem

- Gale's O(n) algorithm for detecting who won a finished game

Ea Ea

- Hex to Y
- Y Reduction
- Sperner's Lemma
- K_5 Embedability?

Topological Dimension Theory

- Covering dimension of the square

## Hex and Probability

Oded Schramm

- Random Turn Hex: At the start of each turn, toss a fair coin. The person who wins the coin plays the next move to the game.
- Percolation probability

[2017-03-04] At long last, the first batch of percolation criticality numbers are in!

```
# one hundred trials on a 7x7 board with coin-toss colouring
:! python HexMonteCarlo.py
[[ 1. 3. 4. 4. 5. 8. 5.]
[ 1. 0. 7. 7. 10. 7. 7.]
[ 4. 6. 4. 10. 10. 12. 7.]
[ 4. 6. 9. 9. 4. 8. 8.]
[ 3. 12. 12. 12. 8. 6. 2.]
[ 4. 9. 7. 7. 9. 5. 4.]
[ 15. 6. 5. 5. 5. 0. 2.]]
# ten thousand trials on a 7x7 board
Simulating 7x7 with N=10000 trials
[[ 128. 232. 363. 478. 582. 704. 1065.]
[ 236. 500. 624. 846. 1001. 1077. 806.]
[ 358. 637. 899. 1076. 1120. 982. 554.]
[ 471. 817. 1114. 1165. 1064. 806. 466.]
[ 572. 1028. 1081. 1086. 908. 636. 350.]
[ 767. 1005. 990. 833. 605. 424. 250.]
[ 1014. 751. 555. 448. 323. 226. 116.]]
```

Monte Carlo simulation

Three variants: Ramsey, Symmetric, and Hex

- Ramsey:
- Symmetric:
- Hex:

## + Programming

Sample RP(x) for small boards. (How big is feasible?)

- Implement Gale
- Toss lots of coins
- Tally the results

Examine the "gradient ascent" for RP(x)

- Fix a gigantic board size (n = 10**6 + 1 to be odd?)
- Pick a position x0 uniformly at random
- Randomly sample RP(x0+eps) for all epsilon neighbours
- Let x1 maximize RP(x-+eps)
- Iterate.
- Keep track of how long it takes x1 to get to xN (a fixed point?).

## Conjecturing

```
Suppose x and y are neighbours.
Q. Is it possible that P(x) = P(y)?
Q. If x is a winning opening and P(x) < P(y) then is y also a winning opening?
Suppose x, y, and z are neighbours.
Q. Do we always get a better "pair of neighbours"?
Can P(x) gradient ascent have periodic points?
What "should" the P(x) flow lines look like?
If x_0 is a winning opening and x_n is its forward orbit under P(x)-ascent
then do we get x_n is a winning opening?
The Role of Central Symmetry P(x) = P(-x)
```