Painting a $3\times3$ Grid

Probability Level 1

Is there any way to paint 3 squares red, 3 squares blue, and 3 squares green in a $3\times 3$ grid so that there are no more than two colors in any row or column?

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1 solution

Geoff Pilling
Apr 21, 2016

The following is one solution:

Follow-up question: Prove that it is the only such coloring, up to permutation of colors and permutation of rows/columns.

Ivan Koswara - 5 years, 1 month ago

Or proving that a color permutation is equivalent to a series of row/column permutations... :)

Geoff Pilling - 5 years, 1 month ago

Well, in general it's false, but in this particular case color permutation is indeed just row/column permutation.

Ivan Koswara - 5 years, 1 month ago

@Ivan Koswara – I wonder... Is it straightforward to calculate how many different solutions there are?

Geoff Pilling - 5 years, 1 month ago

@Geoff Pilling – Oh, whoops, I just realized that permutation of colors even in this case cannot be reproduced by permutation of row/column.

For this particular problem? 72. Fix the red "elbow" (9 choices), fix the two red "legs" (2 choices each), fix the color of the elbow held by the red legs (2 choices).

Ivan Koswara - 5 years, 1 month ago

There is a second solution

R B B

R G R

G G B

Hadi Koteish - 5 years, 1 month ago

I wonder... Can this solution be obtained by some sequence of permuting rows an columns?

Geoff Pilling - 5 years, 1 month ago

Painting a 3 × 3 3\times3 3 × 3 Grid

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Painting a $3\times3$ Grid