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This question is from a popular monthly science magazine in my country:

You have an 8x8 square where any 3 squares forming a tromino (including reflections and rotations) must consist of three different colored squares. At least how many colors do you need to paint the entire board?

Bonus: How about larger boards?

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  • $\begingroup$ Re: the edits, I was sure that the right name was triomino (based on tri(o)-), but I had put the word "popular" right before "science" because I wanted to stress that the magazine deals with popular science, so I don't think it necessarily breaks the adjective order rule. $\endgroup$
    – Nautilus
    Commented Feb 11, 2019 at 14:41

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I think the answer is

5

using the following coloring:

8x8 board

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.

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  • $\begingroup$ I think it's wrong, the angular triomino "2 5 3" (with the 5 at the angle) is repeating. The 2 on first line, 2nd column, and the 2 on 4th line 3rd column, they make the same triomino with the same colours. $\endgroup$
    – Rémi Henry
    Commented Feb 11, 2019 at 10:09
  • $\begingroup$ They are all repeating (since it's a pattern); the point is that each triomino has three different colours. Triominos are not compared with each other (at least that's how I interpreted the puzzle). $\endgroup$
    – Glorfindel
    Commented Feb 11, 2019 at 10:11
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    $\begingroup$ @Oray what's wrong with just extending the pattern? $\endgroup$
    – Glorfindel
    Commented Feb 11, 2019 at 10:39
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    $\begingroup$ If at all possible, it's good to use something else than different shades of red and green for conveying significant information. $\endgroup$
    – Bass
    Commented Feb 11, 2019 at 11:23
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    $\begingroup$ Yeah, good idea. $\endgroup$
    – Glorfindel
    Commented Feb 11, 2019 at 11:31

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