Dilemma
Tic-Tac-Toe
The grid # and the letters X and
O are the same; but make these three changes in the rules:
* Player X starts first, but not
in the center square;
* X and O alternate, until they
fill the grid, or outcome is certain;
* Truce = both XXX and OOO rows;
win/lose = only one sort;
draw = neither XXX nor OOO rows.
Here are some sample games.
Numbers tell order of moves:
X5
| O6 | X3 X5
| O6 | X3 O4 | X3 | O6
O8
| O2 | X9 X7
| O2 | X9 O8 | O2 | X9
X1
| X7 | O4 X1 | O8 | O4 X1 | X7 | X5
Draw Truce X wins
X9
| X3 | O8 X5
| X9 | X7 O6 | X3 | X5
O4
| O2 | O6 O6
| O2 | O8 X7 | O2 | X9
X1
| X7 | X5 X1 | O4 | X3 O4 | X1 | O8
Truce
Truce O wins
O6
| X1 | O4 X9
| X1 | O4 X5 | X7 | X3
X9
| O2 | X3 O8
| O2 | O6 O8 | O2 | X9
X5
| O8 | X7 X5 | X7 | X3 X1 | O6 | O4
Draw Truce X wins
Here is a truce where X started
in the center square:
O4
| X5 | O8
O6
| X1 | X9
O2
| X7 | X3
Compare
the first two games:
X5 | O6 | X3 X5
| O6 | X3
O8
| O2 | X9 X7 | O2 | X9
X1
| X7 | O4 X1 | O8 | O4
Draw Truce
On the second game’s sixth move,
O put down O6 in the top-center square (A2), and then told X, “If you block me
at C2, I’ll block you at B1, and we’ll draw. Better to grab the ABC1 file now,
and let me get ABC2.” X agreed, and they truced. This is classic axial
negotiation.
Dilemma tic-tac-toe is a good
introduction to dilemma games; it is simple, familiar, and a truce under good
axial play.
We can define dilemma tic-tac-toe
on other grids, such as 4x4 (4 in a row) or 3x4 (3 in a row). In the following
diagrams I leave the grid partly blank once the outcome is decided.
A 4 by 4 truce (4 in a row):
X13
| X11 | X9 | X7
X1 | O4 | X3
|
| X5
| O2 |
O12 | O6
| O10 | O8
A
3 by 4 truce (3 in a row):
O10
| O8 | O6
O4 | X1 | X7
X3 | O2 | X9
|
| X5
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