Problem 16 · 2010 Math Kangaroo
Hard
Counting & Probability
careful-counting
A square is split into 4 smaller squares. Each small square is coloured either white or black. How many ways are there to colour the big square? (Two patterns count as the same if one can be turned into the other by a rotation, as shown in the picture.)
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Answer: B — 6
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Hint 1 of 2
Count colourings by how many of the four cells are black: 0,1,2,3,4.
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Hint 2 of 2
Two colourings are the same if a rotation matches them, so merge those.
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Approach: count by number of black cells, up to rotation
- By number of black squares: 0 black gives 1 way; 1 black gives 1; 2 black gives 2 (adjacent or diagonal); 3 black gives 1; 4 black gives 1.
- Total = 1+1+2+1+1 = 6 distinct colourings.
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