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Questions  >>  Olympiad-Math  >>  Class 10
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Q)

What is the minimum number of different colours required to paint he given figure such that no two adjacent regions have the same colour?

$ (A) 3 \\ (B) 4 \\(C)5 \\ (D)6 $

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A)
Solution :
The figure may be labelled as shown.
The regions A, C, E and G can have the same colour say colour 1.
The regions B, D, F and H can have the same colour (but different from colour 1) say colour 2.
The region 1 lies adjacent to each one of the regions A, B, C, D, E, F, G and H and therefore it should have a different colour say colour 3.
The regions J, L and N can have the same colour (different from colour 3) say colour 1.
The regions K, M and O can have the same colour (different fromthe colours 1 and 3). Thus, these regions will have colour 2.
The region P cannot have any of the colours 1 and 2 as it lies adjacent to each one of the regions J, K, L, M, N and O and so it will have colour 3.
The region Q can have any of the colours 1 or 2.
Minimum number of colours required is 3.
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