Problem 1 (3 points). Consider a $2^n \times 2^n$ chessboard with one arbitrary chosen square removed. Prove that any such chessboard can be tiled without gaps by L-shaped pieces, each composed of 3 squares. The following figure shows how to tile a $4 \times 4$ chessboard with the square on the left-top corner removed, using 5 L-shaped pieces.
Added by Encarnacion W.
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Start with a 2 x 2 chessboard with one square composed of 3 squares. This means that one of the squares in the 2 x 2 chessboard is larger than the others. Show more…
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Sri K.
An L-tile is a shape consisting of 3 boxes which form an "L"; for example, the bottom left corner of the chessboard, together with the box immediately above it and the box immediately to the right of it, form an L. If R is a board, a tiling of L is a way to cover R completely with L-tiles such that each tile lies completely on the board and such that no two tiles overlap. Check for yourself that if the board R is a 2 by 3 rectangle, then there are two ways to tile R with L-tiles. Prove that a board R has an L-tiling in the following situations: R is a 2^k by 2^k chessboard with one corner 1 by 1 square removed. (Note that when k = 1, R is just an L-tile.) R is a 2^k by 2^k chessboard with any single 1 by 1 square removed.
Madhur L.
The 3 by 3 square grid has 9 dots equally spaced. How many squares (of all sizes) can you make using four of these dots as vertices? How many such squares exist in a 4 by 4 grid? What about a 5 by 5 grid? How many for an (n+1) by (n+1) grid of dots?
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