Problem 17 · 2016 Math Kangaroo
Stretch
Logic & Word Problems
casework
Peter wants to colour in the cells of a 3×3 square so that every row, every column and both diagonals each have three cells with three different colours. What is the smallest number of colours with which Peter can achieve this?
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Answer: C — 5
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Hint 1 of 3
The centre cell sits on a row, a column, and both diagonals, so four lines pass through it.
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Hint 2 of 3
Look at the centre together with the four corners and ask how many can repeat a colour.
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Hint 3 of 3
Once you see why 3 and 4 colours are forced to clash, find an explicit 5-colour pattern.
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Approach: rule out 3 and 4, then build 5
- Focus on the centre and the four corners: each corner is on a diagonal through the centre, so no corner may match the centre.
- The two main-diagonal corners differ from each other and from the centre, and the same holds for the anti-diagonal corners, so the centre plus four corners already need at least three colours among five awkwardly-linked cells; pushing this through every row, column and diagonal shows 3 colours and then 4 colours always force a repeat somewhere.
- Five colours do work, for example placing colours 1,2,3 / 4,5,1 / 2,3,4-style so every line has three different ones.
- Hence the fewest colours Peter needs is 5.
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