This unit starts out innocently enough. We are just fooling around with puzzles and playing the logic game, Mastermind. It even seems to connect with our previous unit about chance, since the transposition puzzles are related to permutations. But I have a secret agenda. I really want to teach about proof, especially about proving that certain things are impossible. Some of the puzzles we work on are actually impossible to solve and furthermore we can prove that a solution cannot exist. Impossibility proofs often use the method known as "proof by contradiction." This method is introduced in our Mastermind work, and also applied to show that the square root of two cannot be a rational number.
After that, it gets worse. Anyone who hated .999…= 1 or the uncountability of irrational numbers is really going to hate what comes next. Not only can we prove that certain things are possible while other things are impossible, we can also prove that certain things are "undecidable." This is even more abstract.
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