Katie and Andy play the following game: There are \(18\) chocolate bites on a plate. Each player is allowed to take \(1,2\) or \(3\) bites at once. The person who cannot take any more bites loses. Katie starts. Who has the winning strategy?
Arthur and Dan play the following game. There are \(26\) beads on the necklace. Each boy is allowed to take \(1,2,3\) or \(4\) beads at once. The boy who cannot take any more beads loses. Arthur starts - who will win?
Two goblins, Krok and Grok, are playing a game with a pile of gold. Each goblin can take any positive number of coins no larger than \(9\) from the pile. They take moves one after another. There are \(3333\) coins in total, the goblin who takes the last coin wins. Who will win, if Krok goes first?
There are all the numbers from \(1\) to \(2020\) written on the board. Karen and Leon are playing a game where they pick a number off the board and wipe it, together with all of its divisors. Leon goes first - prove that Karen always loses.
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. A player who can’t make a move loses. Who will win, the first or the second player?
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 \(2020\). In how many cases Claire will win?
Fred and Johnny have a number \(1000\) written on the board. In their turn, a player wipes out a number currently on the board and replaces it with either a number \(1\) smaller, or half of the number on the board (rounded down). A player that writes \(0\) on the board wins. Johnny starts, who will win?
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 the 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?