# A Tic-Tac-Toe variant with three marks - winning strategy and chances

We start with a usual $$3 \times 3$$ grid, $$3$$ different marks (say, $$X, O, I$$) and $$2$$ players.

• Player $$A$$ has to make a vertical, horizontal or diagonal line made of $$3$$ identical marks (XXX, OOO or III).

• Player $$B$$ has to make a vertical, horizontal or diagonal line made of $$3$$ different marks (XOI, OXI, XIO, IOX, IXO or OIX).

If both players have a 'winning' position on the last step, the game is a draw.

Below is an illustration (only some separate examples of winning lines are shown for $$A$$ and $$B$$, it's not a full list neither it's an example of a final board). Which player has better chances at winning, if they go first? If the other goes first?

Which player has an easier time forcing a draw in each case?

Are the winning/draw strategies for Players $$A$$ and $$B$$ fundamentally different?

• Can they overlap - can they complete an existing row or column with their own character to win? – Zxyrra Feb 19 '17 at 5:20
• @Zxyrra, would it make the game more interesting? – Yuriy S Feb 19 '17 at 9:35
• potentially although it would make it much harder for A – Zxyrra Feb 19 '17 at 15:02

## Solution

The second player always wins.

## Step-by-step deduction

If A plays first, then

B wins.

Proof: the crucial realisation is that B can always force A to play in a specific square. WLOG, say A's first move is an $X$.

• If A's first move is in a corner (WLOG, an $X$), B plays as follows:

X   O   ?
O   .   .
?   .   .

where the ? marks denote the positions of A's subsequent forced moves (each of which can be either an $X$ or an $O$). Finally, B plays

an $I$ in the centre to guarantee victory.

• If A's first move is in the middle of an edge (WLOG, an $X$), B plays as follows (first $O$, then $I$):

I   .   .
X   O   ?
.   .   .

where the ? mark denotes the position of A's forced move (either an $X$ or an $O$). At this point, A cannot prevent B both from completing the first column and from completing the leading diagonal.

• If A's first move is in the centre (WLOG, an $X$), B plays as follows (first $O$, then $I$):

I   .   .
O   X   ?
.   .   .

where the ? mark denotes the position of A's forced move (either an $X$ or an $O$). At this point, A cannot prevent B both from completing the first column and from completing the leading diagonal.

If B plays first, then

A wins (thanks @IvoBeckers for help with this part of the proof).

Proof: we can more or less adapt the same proof as used in the first half of the solution.

• If B's first move is in the centre or in the middle of an edge, then A can employ a strategy identical to B's strategy described above, except playing $X$ at every turn instead of $O$ or $I$, thus forcing B's hand in the same way as B forced A's when A went first.

• If B's first move is in a corner (WLOG, an $X$), A plays more $X$'s, first in the centre and then adjacent to the filled corner:

X   X   .
.   X   .
.   .   ?

where the ? mark denotes the position of B's forced move (either an $O$ or an $I$). At this point, B cannot prevent A both from completing the first row and from completing the second column.

• I don't get it so far. Why wouldn't A just continue building an X line on his second go? So far B only managed to fill 1 cell, so there are 2 cells available. Then on the next step B makes it impossible to finish the line of course, but A can still try to continue in another cell – Yuriy S Feb 18 '17 at 18:18
• @YuriyS At every stage, A has to play so as to block B from completing a row or column. (To address your specific question of why A doesn't play an X, I'll need to know which game/scenario you're talking about.) – Rand al'Thor Feb 18 '17 at 18:19
• @ randal'thor, you are right, B would just win on their second step in the case I described. Sorry – Yuriy S Feb 18 '17 at 18:22
• To complete your answer: the person that goes second always wins. When B goes first, then for all three examples you have replace all O and I in your examples with X and that's the strategy that player A has – Ivo Beckers Feb 18 '17 at 18:59
• @Ivo I don't think that always works. For instance, take the first example: B plays $X$ in the corner; A plays $X$'s next to that forcing B to play $O$ and $I$ in the next corners; then A can't play $X$ in the middle! – Rand al'Thor Feb 18 '17 at 19:06