A pentomino consists of five unit squares stuck together so that each square shares at least one whole side with another square. There are 12 types of pentominoes. Here is an example:
- Find the other 11 types of pentominoes.
- Find all different rectangles with integer sides greater than 2 whose area equals 60. (Why must the sides be greater than 2?) Show that each of these rectangles can be exactly covered (tiled) by the set of 12 pentominoes (each piece gets used exactly once in covering each square). You are allowed to rotate and reflect the pieces to tile the rectangles.
- Use the 12 pentominoes to form the net of a cube. (A net is a two-dimensional figure that, when cut out and folded up, forms a solid figure. For example, the figure below is the net of a regular tetrahedron. If you cut it out, and fold up the outer triangles, you'll get a tetrahedron.)
- Play a game using a checkerboard and the 12 pentomino pieces (sized appropriately). Two players play by alternately placing one of the pentominoes on the board. The loser is the player who cannot place a piece, either because there are no more pieces or there is no room left on the board. What did you learn playing this game? What is the minimum number of moves the game can last? What's the maximum?
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