Rob Bayer Math 55 Worksheet July 13, 2009 Strong Induction 1. Prove that if n ≥ 18, then you can make n cents out of just 4- and 7-cent stamps 2. Consider a game in which two players alternate turns taking as many stones as they want from one of two piles of stones. The player who removes the last stone wins. (a) Show that if the two piles start with the same number of stones, then the second player can always win. (b) Show that even if the piles start with diﬀerent amounts, the second player can still always win. 3. Here we’ll use well-ordering to show that x 2 + y 2 = 3 xyz has no solutions in positive integers. (a) Show that any solution must have x ≡ 0 (mod 3) ,y ≡ 0 (mod 3). (b) Use well-ordering to show that if there are any solutions, there must be at least one that makes x + y + z minimal. Call it ( x0 ,y0 ,z0 ). (c) Combine parts a,b to show that ( x0 3 , y0 3 ,z0 ) must also be a solution. (d) Explain why this is a contradiction.
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