The numbers from \(1\) to \(2025\) are written on a board. Karen and Leon are playing a game where they pick a number on the board and wipe it, together with all of its divisors. Leon goes first. Show that he has a winning strategy.
Katie and Juan played chess for some time and they got bored - Katie was winning all the time. She decided to make the game easier for Juan and changed the rules a bit. Now, each player makes two usual chess moves at once, and then the other player does the same. (Rules for checks and check-mates are modified accordingly). In the new game, Juan will start first. Show that Katie definitely does not have a winning strategy.
Two players are emptying two drawers full of socks. One drawer has 20 socks and the other has 34 socks. Each player can take any number of socks from one drawer. The player who can’t make a move loses. Assuming the players make no mistakes, will the first or the second player win?
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?