We are taking the unused cards from the top of the deck and putting them on the bottom as we spell the card names, so we can consider the deck as being similar to a keychain with 52 keys on it. Each time we finish spelling out one of the names we take a card out of the loop, but we never change the relative positions in the deck.
The problem can be solved by drawing a circle of 52 squares, unlabeled originally, and then just proceeding around the circle, counting along the squares and filling in the card names as we know where they belong. Note - It's important to keep track of the 'first' square in the circle, as it corresponds to the card which is originally on top of the deck.
So, we count past the first three squares (A-C-E) and label the 4th square with the Ace of Clubs. We then count past 3 more empty squares (T-W-O) and label the next square with the 2 of Clubs. (So, the 2 of Clubs should be in the 8th square along the circle, for those of you following along!) Then count past 5 more empty squares (T-H-R-E-E) and label the next square (the 14th from the start) with the 3 of Clubs. Continue with this procedure until you have labeled the 49th square in the circle with the 10 of Clubs.
At this point, you will have to start going around the circle for the second time. But that's ok; it just means that we have gone through the deck one full time as we perform the card trick - the unused cards have cycled back to where they were originally, and we will begin a second cycle. So, count 4 more empty squares (J-A-C-K) and you will label the next empty square (it's only the 2nd square from the start of the circle!) with the Jack of Clubs.
Now (and here's the clincher), as we start going through the circle for the second time, it's important to remember that all of the labeled squares represent cards that have been REMOVED from the deck at this point. (We are naming the cards and then leaving them face up on the table - or, if you like, we have taken those keys off the keyring.) This means that we have to skip over these squares as we count - in other words, only the unlabeled squares represent cards that are still in the deck, so they are the only ones we count as we continue the procedure.
So, we have placed the Jack of Clubs. Now, we have to count 5 *empty* squares (Q-U-E-E-N), and then label the next empty square with the Queen of Clubs. Try this! You will notice that you have to skip two squares, as they are already labeled with the Ace and Two of Clubs... and the Queen of Clubs ends up being placed in the 10th square from the start of the circle.
At this point, you have everything you need to know to finish filling in all of the squares with the rest of the cards. Just remember, as you count you must skip over the squares that have already been labeled. After you finish the Clubs, you will put in all the Diamonds (Ace to King), then the Hearts, and finally the Spades. Once you have finished, you will hopefully arrive with the following sequence of cards - and can then amaze your friends and family with the card trick!
9S, JC, 5S, AC, KD, KH, 7D, 2C, 6H, QC, 10H, AH, 3S,
3C, 8S, KC, 8D, 7S, 4C, 2H, AD, AS, 7H, 5C, 9D, 2D,
JH, 6C, QS, 6S, 10D, 3H, 3D, 7C, 4S, 2S, 8H, JD, 4D,
8C, JS, 4H, QH, KS, 9C, 5D, 10S, QD, 10C, 9H, 6D, 5H.
-John
The sequence, and one solution method
