Try each problem before reading the solutions. Write down the step that makes your argument complete.
Problems
1. A coin starts heads up. Can exactly 15 flips return it to heads? Explain.
2. There are seven coins, all tails up. A move flips any two distinct coins. Can all seven finish heads up?
3. Twelve coins start tails up. A move flips exactly two distinct coins. Prove which final arrangements are reachable.
Solutions
1. No. Every pair of flips cancels, so 15 flips leave the coin tails up.
2. No. The number of heads changes by −2, 0 or 2. It starts even and stays even, while seven is odd.
3. Exactly the arrangements with an even number of heads are reachable. Parity gives necessity. For sufficiency, pair up the positions that should become heads and flip each pair once.
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