# How many colors does it take?

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?

• 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. – Nautilus Feb 11 '19 at 14:41

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 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. – Rémi Henry Feb 11 '19 at 10:09
• 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). – Glorfindel Feb 11 '19 at 10:11
• @Oray what's wrong with just extending the pattern? – Glorfindel Feb 11 '19 at 10:39
• If at all possible, it's good to use something else than different shades of red and green for conveying significant information. – Bass Feb 11 '19 at 11:23
• Yeah, good idea. – Glorfindel Feb 11 '19 at 11:31