Three-Dimensional Noughts and Crosses: A Combinatorial Problem

Constraints

The following constraints are used in this example:

  • bounds on variables

  • for $(i,j,k) \in \text {CELLS}$,

    \[ \sum _{\text {color} \in \text {COLORS}} \Variable{IsColor[i,j,k,color]} = 1 \]

  • for $\text {color} \in \text {COLORS}$,

    \[ \sum _{(i,j,k) \in \text {CELLS}} \Variable{IsColor[i,j,k,color]} = \Argument{num\_ balls[color]} \]

  • for $\text {line} \in \text {LINES}$ and $\text {color} \in \text {COLORS}$,

    \[ \sum _{(i,j,k) \in \mr{CELLS\_ line[line]}} \Variable{IsColor[i,j,k,color]} - \left|\mr{CELLS\_ line[line]}\right| + 1 \le \Variable{IsMonochromatic[line]} \]