It is impossible to completely tile the plane using only regular pentagons. However, can you identify at least three different types of pentagons (each with at least two different corresponding sides AND angles) that can be used to tile the plane in three distinct ways? Here essentially different means the tilings have different patterns.
Draw how to tile the whole plane with figures, made from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), where squares are used the same number of times in the design of the figure.
Karl and Louie are playing a game. They place action figures around a round table with 24 seats. No two figures are allowed to sit next to each other, regardless of whether they belong to Karl or Louie. The player who cannot place their figure loses the game. Karl goes first - show that Louie can always win.
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 takes a positive number of coins, at most \(9\), from the pile. They take turns one after another. There are \(3333\) coins in total and the goblin who takes the last coin wins. Who will win if Krok goes first?
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?