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I think the answer is

5

using the following coloring:

For other board sizes,

5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)

Reasoning:

Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a triomino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.

I think the answer is

5

using the following coloring:

For other board sizes,

5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)

Reasoning:

Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a tromino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.

I think the answer is

5

using the following coloring:

For other board sizes,

5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)

Reasoning:

Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a triomino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.

I think the answer is

5

using the following coloring:

For other board sizes,

5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)

Reasoning:

Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a triomino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.

I think the answer is

5

using the following coloring:

32415324
15324153
24153241
53241532
41532415
32415324
15324153
24153241

For other board sizes,

5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)

Reasoning:

Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a triomino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.

I think the answer is

5

using the following coloring:

For other board sizes,

5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)

Reasoning:

Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a triomino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.