"Below right is a tile made of three cells in a line. Given that there are 33 cells in the grid, can you cover it with 11 of these tiles? If not, why not? Solution No, you cant. Every position of the tile covers a blue, yellow and red cell. If a covering existed, the grid would have to have 11 cells of each colour. But there are 12 red cells and 10 yellow ones."
"2. Assembly needed The left-hand shape below can be cut into four identical pieces, cutting only along the black lines, which can be fitted back together to make a square, as shown in the right-hand shape. Can you find a different way to cut the left-hand shape into four identical pieces that can be rearranged to make a square? (The pieces may be rotated or reflected.) Solution"
"3. Pizza party Below is one way to divide three pizzas among five people. Three people get a 3/5 slice, and two get a 2/5 and a 1/5 slice. Another way would be to divide each pizza into five equal slices, with each person getting three. What's the smallest number of pieces so that each person gets exactly the same numbers and sizes of pieces? Solution Ten pieces. Each gets a half and a tenth."
A square grid of 33 cells with three corner cells missing cannot be tiled by eleven straight three-cell tiles because each tile covers one blue, one yellow and one red cell, but the grid contains 12 red and 10 yellow cells, not eleven of each. A left-hand shape can be cut along black lines into four identical pieces that reassemble into a square; an alternative dissection into four identical pieces that also form a square is requested. Three pizzas can be divided among five people so that each person receives identical pieces; the minimal solution uses ten pieces, giving each person a half and a tenth.
Read at www.theguardian.com
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