Problems

Tommy and Claire are going to get some number of game tokens tomorrow. They are planning to play a game: each player can take \(1,4\) or \(5\) tokens from the total. The person who can’t take any more loses. Claire will start. They don’t know how many tokens they will get. They might get a number between \(1\) and \(2025\). In how many cases Claire will have a winning strategy?

Fred and Johnny have the number \(1000\) written on a board. Players take turn to wipe out the number currently on the board and replace it with either a number \(1\) smaller, or half of the number on the board (rounded down). The player that writes \(0\) on the board wins. Johnny starts, who has the winning strategy?

You take nine cards out of a standard deck (ace through 9 of hearts), put them all face up on a table and play the following game against another player: Both players take turns choosing a card. The first player to have three cards that add up to 15 wins. The ace counts as one.
If both players play optimally, which player has a winning strategy?

Andy and Melissa are playing a game using a rectangular chocolate bar made of identical square pieces arranged in \(50\) rows and \(20\) columns. A move is to divide the bar into two parts along a division line. Two parts of the bar stay in the game as separate pieces and cannot be rotated, but both can continue to be divided. However, Melissa can only cut along the vertical lines and Andy can only cut along the horizontal lines. Melissa starts. Who will win?

Terry and Janet are playing a game with stones. There are two piles of stones, one has \(m\) stones and the other has \(n\) stones initially. In their turn, a player takes from one pile a positive number of stones that is a multiple of the number of stones in the other pile at that moment. The player who cleans up one of the piles wins. Terry starts - who will win?

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?

Adi and Maxim play a game. There are \(100\) sweets in a bowl, and they each take in turns to take either \(2\), \(3\) or \(4\) sweets. Whoever cannot take any more sweets (since the bowl is empty, or there’s only \(1\) left) loses.

Maxim goes first - who has the winning strategy?

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).