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Weekly Problem 41 - 2007
The Queen of Spades always lies for the whole day or tells the truth for the whole day. Which of these statements can she never say?


Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

We have to prove the statement:

If a regular icosahedron has three red vertices then it has a vertex that has at least two red neighbours.

We can use an argument by contradiction. We suppose that NOvertex has more than one red neighbour and reach a contradiction thus showing that this statement must be false

Without loss of generality you can think about the top vertex being red and decide what that means for the other vertices around it.