Solution for #34

The problem is solved by assuming positions for the ace, determining what the order for the first shuffle would be based on those assumptions, until one such assumption does not produce a conflict. Consider:

We will first assume the ace was in the second slot after the first shuffle. So the shuffling algorithm would always place the card in the first slot into the second slot. Therefore, the 9 must have been the first card after the first shuffle, or it couldn't have ended up as the second card after the second shuffle.

Now that we know that the 9 must have been the first card after the first shuffle, we know that the shuffling algorithm takes the card in the ninth slot and puts it into the first slot. So after the first shuffle, the 10 must have been in the ninth slot, or the 10 would never have ended up as the first card after the second shuffle.

We follow this logic pattern until our knowledge of the order of the cards after the first shuffle is complete. It turns out that this is the correct answer. If we assume the ace to be in any other position other than the second, then we will eventually encounter a contradiction, where two cards must go into the same slot.

So the final ordering is: 9, A, 4, Q, J, 7, 3, 2, 10, 5, K, 8, 6.