Better known as the Down Under deal, it describes a situation where you take a packet of cards of any size, and alternately deal a card (Down) onto the table, and then tuck a card (Under)neath the remaining cards, continuing in this way until only one card remains in your hand.
Clearly, for a fixed-size packet, the card that remains is determined by its initial position in the packet. The following tricks show how this principle can be exploited by some subtle card handling and mental arithmetic done on the fly.
(The term Australian Shuffle we actually reserve for the case where the last remaining card is placed on top of the re-ordered packet in the Down Under deal, thus giving rise to a a well-defined shuffle, or permutation, of the cards.)
We start with a trick found in Martin Gardner's book The Unexpected Hanging and Other Mathematical Diversions (pages 152 and 155--6), where the author reveals that he learnt the basic idea from Mel Stover. He credits John Scarne with the presentation below, mentions a Bob Hummer version from 1939, and describes several variations on the theme due to Sam Schwartz and others.
Hand out the deck for shuffling. Take it back, and shuffle further. Ask a volunteer to call out a random number bigger than 10. Count out that many cards from the deck and hand them to the volunteer, who then does the Down Under deal, under your direction, until just one card remains. You correctly name that card.
You must know the top card of the deck at the outset. This is easy enough to accomplish: for instance, at some point during your shuffling, after the deck has been genuinely randomized by the volunteer, peek at the bottom card and then shuffle it to the top. You can do additional shuffling while retaining that key card at the top.
Next, you need to find twice the difference between the number called out and the highest power of 2 which is strictly less than it. For instance, if 14 is called out, then compute 2 x (14-8) = 12. Now count out that many (12 in this case) cards casually, dealing into a pile, and thus reversing them. Gather the cards while feigning poor memory, saying, "How many did you say? Oh, fourteen,'' and scoop two extra cards off the deck and place them beneath the cards in your hard. Thus you now have the card whose identity is known in position 12 in a packet of 14 cards. If a power of 2, e.g., 16, is called out, then since 2 x (16-8) = 16, simply count out that many cards into a pile, thus reversing them. In either case, hand this packet to the volunteer, and carefully direct the Down Under dealing. The last card is guaranteed to be the original top card.
Everything becomes clearer if we convert to base 2. First suppose the called out number is abc...e in base 2, and is not itself a power of 2. The highest power of 2 less than this is 100...0, which upon subtraction from 1abc...e yields abc...e. Twice that is abc...e0. You count out that number into a pile, thus reversing their order, and finally scoop the remaining cards underneath. This buries the original top card abc...e0 deep in a pile of size 1abc...e. If, on the other hand, a power of two, say 1000...0, is called out, then the dealing places the original top card at the bottom of the pile. The reader can now check that in both cases, the Down Under deal results in the known card being the last card remaining in the hand.
John Scarne also turned the basic idea of the last trick into a brilliant effect where the tables are turned: the volunteer handles the cards throughout and finds a card that you pick!
The deck is freely shuffled. You announce that you are merely going to think of a card, and see if the volunteer can locate it. Have a random number greater than 20 called out, and ask the volunteer to deal that many cards face up into a pile, indicating that you are going to see if your mentally chosen card is among those. At the conclusion of the deal, claim that you did indeed see your card among those counted out, and have the remainder of the cards put away. As an apparant afterthought, write the name of your card on a piece of paper, and have that folded and put to one side. The cards dealt are picked up and placed face-down again, and the volunteer does the Down Under deal until just one card remains. You remark, "If that's the card I thought of, you're a better magician than I." It does indeed turn out to be your card, as the piece of paper confirms!
What really happens here is that you use the called-out number to compute the position of the card which will "rise to the top" when the volunteer does the Down Under deal; that's the one you (belatedly) "think of," and write on the paper. To achieve this end, find twice the difference between the number called out and the highest power of two which is strictly less than it. For instance, if 25 is called out, then compute 2 x (25-16) = 18. The 18th card dealt face-up, out of 25, is the one you claim to have thought of in the first place. The cards are then turned face-down again, and the Down Under deal takes care of the rest.
If 32 were called out, then since 2 x (32-16) = 32, the conspicuous last card dealt would be your card. This is too risky, as the volunteer or another audience member may remember later that this was the last card dealt. In such cases, suggest that the packet be shuffled some more, and then you do whatever you have to, to get a peek at the new bottom card before the dealing begins.
When performing any such trick, one soon discovers that if the called-out number is too big, the dealing becomes tedious. Smaller packets work well in any case, say of 10 to 20 cards. A quick examination of the numbers reveals something else of interest, as we shall shortly see, and points to the desirability of a less "top down" card management style. We suggest a new approach to both the mental arithmetic involved and the (false) counting, by presenting such a variation of the original trick. We leave it to the interested reader to make the appropriate adjustments to Scarne's ingenious role reversal treatment.
You must know the top card on the deck at the outset. As seen earlier, if we can get that card to a certain position in the packet, then by means of a mental calculation and a false count, the Down Under deal ensures that the final card in your hand will be the original top card. The work can be simplified quite a bit by considering the following table, which shows what's really going on for packets of size 5 to 16:
The first row lists the size of the packet which will be used, which is the called-out number. The second row lists the positions in the packet, starting from the top, to which the original top card must be moved, computed as in the earlier tricks. The third row lists same these positions, counting from the bottom of the packet. If these are added to the corresponding numbers in the first row, it becomes clear how to find them with less effort in the first place.
Simply subtract the packet size from 9, or 17 or 33, whichever is closer, so as to get the smallest possible positive result.
E.g., suppose, that 14 is called out, so that we wish to get the top card of the deck to position 12 from the top. This is the same as getting it to position 3 from the bottom. Certainly it is easier to subtract 14 from 17 to get 3 than it is to do the 2 x (14-8) = 12 computation suggested earlier. Even better, there is a more natural way to do the false count: simply count out 3 cards casually from hand to hand (this does not reverse them) and drop them, as a unit, in a pile on the table, and then count out the remainder (``Four, five, six, ..., fourteen'') and drop those on top of the first pile.
The two actions are identical, most audiences will attach no significance to the break in the count. You can even take more breaks, if you wish to muddy the waters further. For instance, you could peel off small groups of cards and drop them on the table casually, as long as the first group contains exactly 3 cards. Thanks to Joe M. Turner of the Georgia Magic Club (I.B.M. Ring 9) for suggesting this false count variation.
You now have the card whose identity you know in position 3 from the bottom, or 12 from the top, in a packet of 14 cards. As before, hand this packet to the volunteer, and carefully direct the Down Under dealing. The last card is guaranteed to be the original top card.
Note that if 17 is called out, then we subtract it from 33, not from itself!
The moral? Sometimes, when looking up from the bottom of a well, you can see better than you can peering into it from the top; once your eyes adjust.
Colm Mulcahy (firstname.lastname@example.org) has taught at Spelman College since 1988. He is currently the chair of the department of mathematics there. He wants readers to know that "Scarne" has two syllables. You can find more of his writing on mathematical card tricks at http://www5.spelman.edu/~colm/cards.html.