Problems

Anna and Beth played rock paper scissors ten times. Rock beat scissors, scissors beat paper and paper beat rock. Anna used rock three times, scissors six times and paper once. Beth used rock twice, scissors four times and paper four times. None of the ten games was a tie. Who won more games?

Consider the \(4!\) possible permutations of the numbers \(1,2,3,4\). Which of those permutations keep the expression \(x_1x_2+x_3x_4\) the same?

In the picture below, there are the \(12\) pentominoes. Is it possible to tile a \(6\times10\) rectangle with them?

Show how to tile a \(5\times12\) rectangle with the twelve pentominoes.

Show how to tile a \(4\times15\) rectangle with the twelve pentominoes.

Is it possible to tile a \(3\times20\) rectangle with the twelve pentominoes?

Show to how to cover the plane with this cube net:

Show how to tile the plane with this cube net:

Four football teams play in a tournament. There’s the Ulams (\(U\)), the Vandermondes (\(V\)), the Wittgensteins (\(W\)) and the Xenos (\(X\)). Each team plays every other team exactly once, and matches can end in a draw.
If a game ends in a draw, then both teams get \(1\) point. Otherwise, the winning team gets \(3\) points and the losing team gets \(0\) points. At the end of the tournament, the teams have the following points totals: \(U\) has \(7\), \(V\) has \(4\), \(W\) has \(3\) and \(X\) has \(2\).

Work out the results of each match, including showing that there’s no other way the results could have played out.

Naomi and Rory get tired of playing Nim, so decide to change the rules to mix it up. They call their new variant ‘Wonim’. There are two piles of four matchsticks each. They take it in turns to take matchsticks. Each player has to take at least one matchstick, and they can take as many as they like from one pile only.

Except, their new rule is that a player cannot take the same number of matchsticks that their opponent just did. For example, consider Wonim(\(5\),\(10\)). If Naomi’s first move is to take \(4\) matchsticks from the pile of size \(5\), turning the game to Wonim(\(1\),\(10\)), then Rory cannot take \(4\) matchsticks - he has to take more or less. A player loses if they cannot go - this can happen if there are no matchsticks left, or if there are matchsticks left, but they can’t take any since their opponent took that number. e.g. Wonim(\(1\),\(1\)), Naomi takes \(1\), Rory faces Wonim(\(1\)) but can’t move since he’s not allowed to take \(1\).

In the game Wonim(\(4\),\(4\)) with Naomi going first, who has the winning strategy?