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: Lift every second cup. If you see a tails, turn all tails to heads and win. Otherwise, turn all heads to tails and obtain .
- Turn 4: Flip four consecutive coins. From the position of at least one heads, lift every second cup. Turn all to tails. This gives .
- Turn 5: Flip every second pair of cups and turn all to tails. This either wins the game or leaves .
- Turn 6: Flip every second cup. If you find the remaining heads, you win. Otherwise flip all to heads giving you .
- Turn 7: Flip every second pair. If all you see are "ht" or "th" combinations, flip all tails to heads giving . If, on the other hand, you find the "hh" combination, flip the "h" corresponding to the other tails to tails. This leaves . In this case, skip to Turn 11.
- Turn 8: We are now in the situation . Flip over 4 consecutive cups. If you see a tails, flip over every fourth cup from the tails and turn all to heads. This is possible provided m≥4. In this case, you win. If you only see only heads, keep flipping over consecutive cups until you find the first tails. Flip the offending heads (4 off the tails) to tails. This gives .
- Turn 9: Flip every second cup. If you see a tails, flip them to heads and win. Otherwise, flip every second revealed heads to tails. This yields .
- Turn 10: Flip every second pair. If you see all heads or all tails, you win. Otherwise, flip all coins to obtain .
- Turn 11: Turn every second cup and flip all coins.