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Proximity

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Paradoxical

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Statements A, B and E may be made by the Queen of Spades, whether it is a day on which she is telling the truth or a day on which she is lying. She may also make statement C providing she is telling the truth that day. However, if she is telling the truth on a particular day then she could not make statement D since that would be a lie. Also, if she is not telling the truth on that day then she could not make statement D either, since that would then be a true statement. So she cannot make statement D.

This problem is taken from the UKMT Mathematical Challenges.
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