# What is the formula for winning at Pentago?

1. Nov 6, 2015

### greswd

Pentago is a board game and you can think of it as a highly advanced version of tic-tac-toe.

With the aid of supercomputers, it has been strongly solved. Just like tic-tac-toe, it is possible for the player who starts first to always in.

I'm looking for a formula to always win at Pentago if I'm the first player. For tic-tac-toe, always mark the central square. For Pentago, never touch the 4 corners.

Tic-tac-toe is simple enough, but what is a formula for Pentago that can be applied by humans anytime?

Just like once someone memorises the algorithm, he can solve any Rubik's cube problem.

2. Nov 7, 2015

### Orodruin

Staff Emeritus
Tic-tac-toe is impossible to lose for either player unless they make mistakes. If both players know what they are doing, it will end in a draw.

3. Nov 15, 2015

### greswd

So in Tic-Tac-Toe perfect play, it will always end in a draw. What constitutes perfect play for Pentago?

4. Nov 15, 2015

### willem2

look here:

https://perfect-pentago.net/

They store the result of all positions with less than 18 stones. Every first move is winning, except for the corners.

5. Nov 15, 2015

### greswd

omg i didnt see that earlier. looks like they've just updated the site

6. Nov 15, 2015

### greswd

but anyway, is there a general formula that can be applied to all stages of the game? what constitutes perfect play?

7. Nov 16, 2015

### jbriggs444

Depends on what you mean by "formula".

Is there an algorithm to determine perfect play? Yes.

Is there a polynomial formula that returns an optimal move for any possible position. Yes, trivially. Encode the board position as 36 variables x1 through x36 with values 0 (no stone), 1 (black stone) or -1 (white stone). Then a polynomial with 336 terms exists which will return the required result. That polynomial could, in principle, be constructed as a sum of 336 products like the following one that embodies the starting board position:

k * (x1 - 1 )(x1+1)(x2-1)(x2+1) ... (x36-1)(x36+1)

Is there a formula that a human can use in reasonable time with pencil and paper? The fact that the Perfect Pentago site used a supercomputer to accomplish the task suggests that no such formula is known.

8. Nov 16, 2015

### greswd

perhaps it could be simplified? that would be very hard though