We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
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?
A: "Yesterday, I told the truth";
B: "Yesterday, I lied";
C: "Today, I tell the truth";
D: "Today, I lie";
E: "Tomorrow, I shall tell the truth".
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
This problem is taken from the UKMT Mathematical Challenges.