Problems
Michelle and Mondo play the following game, with Michelle going first. They start with a regular polygon, and take it in turns to move. A move is to pick two non-adjacent points in one polygon, connect them, and split that polygon into two new polygons. A player wins if their opponent cannot move - which happens if there are only triangles left. See the diagram below for an example game with a pentagon. Prove that Michelle has the winning strategy if they start with a decagon (\(10\)-sided polygon).
One square is coloured red at random on an \(8\times8\) grid. Show that no matter where this red square is, you can cover the remaining \(63\) squares with \(21\) ‘L’ triominoes, with no gaps or overlaps.
Let \(n\) be a positive integer. Show that \(1+3+3^2+...+3^{n-1}+3^n=\frac{3^{n+1}-1}{2}\).
Show that all integers greater than or equal to \(8\) can be written as a sum of some \(3\)s and \(5\)s. e.g. \(11=3+3+5\). Note that there’s no way to write \(7\) in such a way.
Find a formula for \(R(2,k)\), where \(k\) is a natural number.
Show that \(R(4,3)\ge9\). That is, there exists a way of colouring the edges of \(K_8\) with no red \(K_4\), nor any blue \(K_3\).
Show that \(R(4,4)\ge18\) - that is, there’s a way of colouring the edges of \(K_{17}\) such that there’s no monochromatic \(K_4\).
Show that \(R(4,3)\le9\). That is, no matter how you colour the edge of \(K_9\), there must be a red \(K_4\) or a blue \(K_3\).
Show that \(R(4,4)\le18\) - that is, no matter how you colour the edges of \(K_{18}\), there must be a monochromatic \(K_4\).