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 5-by-5 Square Tic-Tac-Toe (Posted on 2012-01-26)
My brother and I used to play a game we created called 5-by-5 Square Tic-Tac-Toe. The game is played on a 5-by-5 grid. The players place their respective symbol (either X or O) in an available square. The object of the game, rather than to get three-in-a-row like in Classic Tic-Tac-Toe, is to create a square using 4 of one's own symbols.

Here's an example to help demonstrate some strategies:

```+ - + - + - + - + - +
|   |   |   |   |   | 5
+ - + - + - + - + - +
|   |   | O | X |   | 4
+ - + - + - + - + - +
|   |   | X |   |   | 3
+ - + - + - + - + - +
|   | O |   | X |   | 2
+ - + - + - + - + - +
|   |   |   |   |   | 1
+ - + - + - + - + - +
A   B   C   D   E```

Here, after X's third move, O must play E3 to block the square X is threatening. X must then play D1 to block O's square.

```+ - + - + - + - + - +
|   |   |   |   |   | 5
+ - + - + - + - + - +
|   |   | O | X |   | 4
+ - + - + - + - + - +
|   |   | X |   | O | 3
+ - + - + - + - + - +
|   | O |   | X |   | 2
+ - + - + - + - + - +
|   |   |   | X |   | 1
+ - + - + - + - + - +
A   B   C   D   E```

At this point, O can not threaten another square, so her best strategy is to block as best she can. Suppose she plays C2, which would mean she can block two of X's possible threats (D3 and C1), and has the additional advantage of offering three potential threats for her next move: B1 (threatens C1, and vice versa, noted below with G), A4 (threatens A2, noted with I ), and B4 (threatens D5, noted with H ).

```+ - + - + - + - + - +
|   |   |   | H |   | 5
+ - + - + - + - + - +
| I | H | O | X |   | 4
+ - + - + - + - + - +
|   |   | X |   | O | 3
+ - + - + - + - + - +
| I | O | O | X |   | 2
+ - + - + - + - + - +
|   | G | G | X |   | 1
+ - + - + - + - + - +
A   B   C   D   E```

The game plays out with X playing E2, O blocking with E1, and X plays B3. O blocks with C1, X counter-blocks with B1, and O blocks with D3 (It might help you to draw this out on a piece of paper, one move at a time). Finally, X plays A4.

```+ - + - + - + - + - +
|   |   |   |   |   | 5
+ - + - + - + - + - +
| X |   | O | X |   | 4
+ - + - + - + - + - +
|   | X | X | O | O | 3
+ - + - + - + - + - +
|   | O | O | X | X | 2
+ - + - + - + - + - +
|   | X | O | X | O | 1
+ - + - + - + - + - +
A   B   C   D   E```

O is unable to block both A1 and D5, so X wins on his next move.

```+ - + - + - + - + - +
|   |   |   | X |   | 5
+ - + - + - + - + - +
| X |   | O | X |   | 4
+ - + - + - + - + - +
|   | X | X | O | O | 3
+ - + - + - + - + - +
|   | O | O | X | X | 2
+ - + - + - + - + - +
| O | X | O | X | O | 1
+ - + - + - + - + - +
A   B   C   D   E```

My brother refuses to play any more, because he believes whoever plays first always wins, assuming everyone plays with optimal strategy and makes no mistakes.

Is my brother right? Can the player who goes first always win?

 No Solution Yet Submitted by Dustin No Rating

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 Thoughts | Comment 1 of 2
I've played this game many many times with family and friends.  I don't have a proof, only a lot of experience.  I therefore feel comfortable working on this problem with everyone.

This grid shows how many squares can be made by using each individual gridsquare.

`+ - + - + - + - + - +| 4 | 7 | 8 | 7 | 4 | 5+ - + - + - + - + - +| 7 |10 |11 |10 | 7 | 4+ - + - + - + - + - +| 8 |11 |12 |11 | 8 | 3+ - + - + - + - + - +| 7 |10 |11 |10 | 7 | 2+ - + - + - + - + - +| 4 | 7 | 8 | 7 | 4 | 1+ - + - + - + - + - +  A   B   C   D   E`

It seems the center gridsquare, C3, is the best starting move.  Anyone else have thoughts?

Edited on January 27, 2012, 4:10 am
 Posted by Dustin on 2012-01-27 04:10:09

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