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., .
Let us consider .
- Turn 1: Flipping every second coin to heads gives .
- Turn 2: Flipping every second pair to heads gives .
- Turn 3: Flip four consecutive cups and eliminate any tails leaving .
- Turn 4: Flip every second cup. If you find the remaining tail, you win. Otherwise flip all to tails giving .
- Turn 5: Flip a cup If you see a heads, flip every second cup and turn all heads to tails. Otherwise, reveal the coins to the left, right, and 3 to the right. If you found another tails, you know which tails to flip to produce and skip to Turn 9. Otherwise, flip the single tails to heads and the heads to the right to tails. This gives .
- Turn 6: Reveal four adjacent coins. This tells you exactly where you are in the sequence. This allows you to either produce (a) (skip to Turn 8) or (b) .
- Turn 7: In case (b), reveal an adjacent set of 3. If you see "tth", "ttt", or "htt", you know which fourth cup to turn and which two coins to flip to make . Otherwise, you are unlucky and found "tht". Flip the second tails to heads and obtain .
- Turn 8: Flip every second pair and you either win or produce .
- Turn 9: Flip every second coin.