We have a deck of \(13\) cards from Ace to King. Let Ace be the first card, \(2\) the second card and so on with King being the thirteenth card. How can you swap \(4\) and \(7\) (and leave all other cards where they are) by only switching adjacent pairs of cards?
How many permutations of 13 cards leaves the third card where it started?
Does there exist an irreducible tiling with \(1\times2\) rectangles of a \(6\times 6\) rectangle?
Irreducibly tile a floor with \(1\times2\) tiles in a room that is a \(6\times8\) rectangle.
Four different digits are given. We use each of them exactly once to construct the largest possible four-digit number. We also use each of them exactly once to construct the smallest possible four-digit number which does not start with \(0\). If the sum of these two numbers is \(10477\), what are the given digits?
Two players are playing a game with a heap of \(100\) rocks, and they take turns removing rocks from the heap. The rules are the following: the first player takes one rock, the second can take either one or two rocks, then the first player can take one, two or three rocks, then the second can take \(1\), \(2\), \(3\) or \(4\) rocks from the pile and so on. That is, on each turn, the players have one more option for the number of rocks that they can take. The one who takes the last rock wins. Who has the winning strategy?
Prove that one can tile the whole plane without spaces and overlaps, using any non self-intersecting quadrilaterals of the same shape. Note: quadrilaterals might not be convex.
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.