![]() However, this rather crude count of the number of clock puzzles ignores the fact that some clock puzzles have no solution. There are thus distinct clock puzzles with N positions, which grows very quickly with N – its values for N = 1, 2, 3, … are given by the sequence 0, 1, 1, 16, 32, 729, 2187, 65536, 262144, … ( A206344 in the OEIS). As mentioned earlier, a clock puzzle with N positions has an integer in the interval in each of the positions. Let’s work on determining how many different clock puzzles there are of a given size. ![]() ![]() We have now selected each position exactly once, so we are done – we solved the puzzle! In fact, this is the unique solution for the given puzzle. We continue on in this way, going through the N = 6 positions in the order 1 – 0 – 3 – 4 – 2 – 5, as in the following image: Three moves either clockwise or counter-clockwise from here both give the 1 in position 3, so that is our only possible next choice. If we start by choosing the 1 in position 1, then we have the option of choosing the 3 in either position 0 or 2. To demonstrate the rules in action, consider the following simple example with N = 6 (I have labelled the six positions 0 – 5 in blue for easy reference): You win the game if you choose each of the N positions exactly once, and you lose the game otherwise (if you are forced to choose the same position twice, or equivalently if there is a position that you have not chosen after performing step 2 a total of N-1 times). During the game, N ranges from 5 to 13, though N could theoretically be as large as we like. Repeat step 2 until you have performed it N-1 times.Update the value of M to be the number in the new position that you chose. You now have the option of picking either the number M positions clockwise from your last choice, or M positions counter-clockwise from your last choice.The user may start by picking any of the N positions on the circle.A challenging late-game clock puzzle with N = 12
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