A Game of Cups and Coins
We will use the following notation: h = heads, t = tails, x = unknown, R=revealed, C=conceiled. Repeating patterns will be denoted in the form . This is a pattern of k values repeated r times. E.g., .
- Turn 1: Reveal n-2 adjacent [R...RCC] and turn all to heads. This yields [h...hxx].
- Turn 2: Reveal n-3 adjacent and 1 singleton [R...RCRC] and turn all to heads. If xx was contained in R...R you win (requires n-3≥2, i.e., n≥6 since n is even). Otherwise the singleton will be one of the x's. Leaving [h...hx]. If the game is not won, we are thus in the situation of [h...ht].
- Turn 3: Reveal n-3 adjacent and 1 singleton [R...RCRC]. If you see the t, flip it and win. Otherwise the singleton is adjacent to t. Flipping the singleton yields [h...htt]
- Turn 4: Reveal two blocks of adjacents, i.e., . If you see both t's flip them and win. Otherwise you see one t and the other t is the adjecent C. The opposite C is h. Flip all coins with even distance to conceiled t to t and all other coins to h. This yields .
- Turn 5: Reveal and flip all. This wins the game and requires n/2≤n-2, i.e., n≥4.