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20 21 mathematIcal domInos proofs by induction Setting up a number of dominos in a row, one domino for each natural number, lets make sure that IF domino n topples, THEN so does domino n 1. If we now topple the first domino, we can be sure that every domino will eventually topple over. Proof by induction is the mathematical counterpart of this insight. Now, instead of the dominos, we have an infinite number of statements, one for each natural number. Here, we can be sure that all statements are true IF we can prove that the first statement is true AND the truth of statement n implies the truth of statement n 1. The first three rows of diagrams opposite show how the first three statements corresponding to the following theorem imply each other Theorem Every 2 n by 2 n board that has been dented in one of its unit squares can be tiled with Lshapes made up of three unit squares. Proof by induction Since a dented 2 by 2 board is an Lshape opposite, top the theorem is true for n 1. Assuming that statement n is true and considering an arbitrary dented 2n1 by 2n1 board, we quarter it and remove three middle squares to create four dented 2 n by 2 n boards center, opposite. By assumption these four boards can be tiled, and the four tilings extend to one of the 2n1 by 2n1 board. Q.E.D. Some of the other tumbling patterns used in the art of domino toppling also translate into methods of proof. In the triangular pattern lower, opposite, for example, the front piece topples all other pieces. The corresponding method of proof can be used to show that Pascals triangle, named after Blaise Pascal 1623 1662, is made up of binomial coefficients see too pages 42 54.
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