# What is the formula for winning at Pentago?

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 win.

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 winning Pentago that can be applied by humans anytime?

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

• I don't think many of us are familiar to Pentago, but personally, I developed an algorithm to always win at Tic-Tac-Toe. Contrary to popular belief, it is ridiculously more easy to win if you start at a corner. You can give it a try and maybe apply the same concepts to Pentago. Nov 7, 2015 at 5:50
• @zickens: If your algorithm always wins at Tic-Tac-Toe, then your algorithm is invalid. The first player can only force a draw unless the second player does the wrong thing, and even then most people learn quickly the optimal strategy. Nov 7, 2015 at 6:35
• The thing is that Tic-Tac-Toe is very limited; the amount of moves possible is very small. Even starting from the center there is a set of moves that will always make the game a draw. The point of my algorithm is to beat the user to its own mistakes; in essence, winning once and drawing all the other games is still winning. If you don't believe me, i could play with you :) Nov 7, 2015 at 7:41

Here's an interactive game where you can see where to go to win, draw, or lose. Unfortunately, it doesn't tell you the optimal move to win as fast as possible, or to lose as late as possible as player 2. I'd really like to see a list of moves sorted by how soon you'd win/lose.

A good strategy seems to be to go in the center of as many squares as possible, then for the places adjacent to your center-held spots, trying to get a line of 3 on a single square. You force your opponent to play defensive the entire time.

• To answer OP's question, the algorithm used by the website is the usual one: to check every possible game. (taking advantage of symmetries etc etc) Jul 29, 2017 at 4:13
• @GPhys That can't be done by a human though, only a computer. "Unlike chess and go, pentago is small enough for a computer to play perfectly: with symmetries removed, there are a mere 3,009,081,623,421,558 (3e15) possible positions." [source] No human can memorize that many. Jul 30, 2017 at 3:51
• The OP's question was about strategies for a person to memorize. Treat it like Chess. Memorize the first few moves, the no-no's, and find some good openings. Jul 30, 2017 at 3:53

I believe I have found the optimal strategy. It wins for white every time, within the first 5 moves. As far as I know, it is unstoppable (if you can combat it please say I would love to know).

Glossary:

• block - 3x3 rotating unit
• corner - the top left, top right, bottom left or bottom right of a block
• side - the left, right, top or bottom of a block

Moves for white)

1. place on side of any block
2. place on any free side of the same block that is adjacent to (1)
3. place on side of the block diagonally opposite to (1) and (2)
4. place on any free side of the same block that is adjacent to (3)
5. place on any corner of the other 2 blocks

(With suitable rotation - you win)

It should look something like this:

• 0X0000
• 00X000
• 000X00
• 0000X0
• 00000X
• 000000

I hope this is clear and I have understood the rules of the game