(by Colm Mulcahy)
Out Of This Whirled (The Odds Are Not Even)
Take a deck of blue-backed cards from its box, and casually shuffle in hand while announcing what is going to happen next. "I need somebody with a really good colour sense, Molly, would you mind?"
Fan the cards, drawing attention to the fact that they are not all face down as is customary, before saying, "I want these to be really jumbled, and not facing the same way which is too predictable. After I've finished mixing further, I'd like you to go through them and deal them into two piles, based only on what you can see. But I don't want you to deal into piles of blue backed face downs and white face ups, anyone could do that. I want you to deal into piles depending on whether you believe that the suits on the card faces are Red or Black. It's easy for face up cards, but for the face down ones you'll have to use your intuition.
Let that sink in, before adding pointedly, "What are the odds that you can do this correctly? For most people the odds would be even, but in your case I'm hoping for a better outcome."
You shuffle the cards in hand a bit, then split the deck into two, combining the halves in a rosette shuffle as explained below. Say, "Your job, Molly, is to tease some pattern out of this whirled mess. Finally, square up the deck and hand it to her, asking to first go through the cards and pull out the Aces. You place one Black and one Red Ace on the table, from left to right, to form leader cards for two piles which will now be dealt.
Point to the two Aces, saying, "I want you to start dealing now, from the top. If you think–or can plainly see–that a card is Black, put it next to this Ace on the left, otherwise put it next to the Red one there." Stop Molly after she has dealt about a third of the deck, and the card on the top of the remainder of the packet in her hands is face down. Take the undealt cards from her and place them beside the card case.
"Let's take a peek at how it's going," you casually remark. Spread out one of the two dealt piles a little, and turn over two or three of the face down cards. If the card is the same colour as the leading Ace, congratulate Molly, and if not, just say, "Oops, nobody's perfect. It's not easy the first time." Replace those cards on the top, once more face down. Repeat for the other pile. Straighten up both piles and push them to one side. [For later reference, we call these piles A and B.] As an afterthought, extract their leading Aces and set them side by side, face up, as leader cards for two new piles, only this time with the Red one on the left.
Have Molly pick up the unused two thirds of the deck beside the card case, and continue as before, until she only has about 15 cards left in her hand. Take those from her, and set them next to the card case. Once again, inspect the dealt piles [which we'll call C and D], turning over two or three of the face down cards in each, and commenting on the success rate so far. Replace those cards on the tops of their respective piles, face down. Straighen up piles C and D, and push them to the side, next to piles A and B.
"Take a break, Molly, you've done well. As a control, let's see how somebody without Molly's special talents would fare. Mohammed, would you like to try?" Now, for the first time, pick up the other two Aces and set them on the table as new leader cards. Have Mohammed deal out into two piles the remaining third of the deck by the card case, again based on guessing which of the face down cards are Black and which are Red.
When Mohammed is finished, have all of the cards he dealt inspected. Like Molly, he'll have had some hits and some misses. Thank him for his participation and turn back to Molly, "Mohammed got some right and some wrong, on average it looks like he did no better than chance would suggest. Molly, we know from peeking earlier that you got a few wrong, but I have a feeling that overall you did much better than Mohammed.
Combine the two Black piles from Molly's attempt, namely piles B and C, and also the two Red piles, namely piles A and D, pulling out the two Aces and setting them on the table face down. "Let's see," you say, fanning one of the resulting two packets of a dozen cards. The face up cards will of course all be the same colour, let's say Black. Now flip this packet over as the tension builds. Alas, not all of the newly visible card faces will be Black, in fact only about half of them will be. Repeat with the supposedly Red packet, with the same results.
Look crestfallen, and step back. "This is not what I was hoping for. I felt sure that Molly had some special intuition when separating the cards. All I asked for was that she use that sixth sense to divide the deck into Blacks and Reds. Looking at the two packets she came up with, however, I don't see that separation."
Spread the cards from one of the packets out again, then perk up and excitedly spread the other packet also. Now look through Mohammed's discarded cards, as you say, "But wait, I've just noticed something! You may recall I said that for most people, when it comes to separating the cards, the odds would be even. Even odds? What was I thinking of? The odds most certainly are not even, and look, what Molly really did was separate the deck into odds, in this pile, and evens in that one. See for yourself. Compare with Mohammed's two piles, which are completely mixed up, there is no separating by parity there. Molly really does have special powers, which even she didn't know about. How odd. Congrats!"
The deck must be prepared in advance, so that when it is taken from its case at performance time, it can withstand the assorted shuffles it is then subjected to. Start by separating the deck into five groups: the Aces, the remaining Black odds (i.e., 3, 5, ..., King), the Red odds, the Black evens (2, 4, ..., Queen) and the Red evens. Each of these groups should consist of a mix of face up and face down cards. Next, shuffle the two odd piles together, to get a good mix of Blacks and Reds, and do the same with the two even piles. Split both of these packets of twenty-four cards in half, making sure that the bottom card in each odd pile is a face down Red King, and the bottom card in each odd pile is a face down Black 8, and recombine to form two new packets of twenty-four cards, each with the odds on top and the evens on the bottom. Finally, reassemble the deck by putting one of these packets on top of the other, and inserting the four Aces anywhere, but not all facing the same way. Replace the deck in its box.
When the deck is later removed from its box, mentally designate as "the bottom" the end with the visible Black 8. Grip the top of the deck firmly, and shuffle off cards from the bottom fifth of the deck, being careful always to replace the top each time. The entire packet can be flipped over at will, interchanging the roles of top and bottom, and this kind of shuffling done to either end. Ignoring the Aces, the current of play, unknown to your audience, is that the deck still consists of layered odds and evens, and the middle cards have not moved at all.
Having commented that the cards are quite mixed up, you now fan the deck quickly to prove the point, while secretly looking for the Black 8 at or near the middle. You may have to flip the whole deck over to find it; it may not be at the exact middle due to the Ace placements from earlier. Split the deck at this point, so that in each hand you now have approximate half packets consisting of: odds on top, evens on the bottom.
Now, you perform Lennart Green's rosette shuffle, which is equivalent to a riffle shuffle. With the piles side by side, use the fingers to "twirl" the left pile into a rosette, repeating for the right pile, and then pushing the rosettes together. These image show the sequence for a face down deck. Finally, square up the packet.
As you do this, you may wish to comment, "I'm going to whirl the two halves of the deck like this, and then push them together to get a big whirled mess." At the conclusion of this you square up the packet, and do a little more controlled shuffling; this time of up to a third of the deck at either end.
Furthermore, as explained in the Low-Down Dealing Without the Dealing section here last time around (April 2011), you can throw in a new type of card mixing here which only seems reasonable precisely because the cards do not all face the same way: you can lift or thumb off up to a third of the deck from either end, and replace it flipped over as a unit. (In the Card Colm referred to above we suggested flipping up to half of the cards each time, but that's too risky here, as the middle part of the deck likely to be no longer clearly split into odds and evens.)
Do this several times, alternating which end of the deck is "up," you may even wish to fan the cards to be flipped each time, remarking, "Molly, it would really help you in a few minutes if you knew which of these cards have Black faces and which have Red ones. Watch carefully while I thumb off clumps here, and memorize the colour of each card face." For comedic effect, speed up as you repeatedly flip clumps as well as the entire deck.
The effect now proceeds as described earlier, with a few key twists.
First, when you stop Molly about a third of the way through the deck and take the remaining cards from her, the real goal is to sneakily flip those over so that when she resumes she is dealing from the other end of the deck, leaving Mohammed in the lurch with the original middle third, which won't be perfectly odd/even separated owing to the rosette shuffle. To make this less flipping obvious, we recommend ensuring that at each "pause point" the cards at both ends face inwards, so that each end of the packet is visibly capped with a Blue card back.
Second, while Mohammed is doing his thing, you discreetly move the four set aside piles out further to the side. In fact, you do this in such a way piles B and D are switched. Then, when you claim to be combining the two Black piles (A and D) you are really combining piles A and B, which are both odd. Similarly, when you claim to be combining the two Red piles (B and C) you are really combining piles D and C, which are both even.
The trick can be repeated, if the deck is given out for shuffling, mixing face up and face down cards, before being replaced in its case and that put in your pocket. A second, identical looking, deck, is now removed from the same pocket and a second volunteer sought. Only this time, the set up is not based around odd versus even, but rather, prime (2, 3, 5, 7, Jack, King) versus composite (4, 6, 8, 9, 10, Queen) values. Those on the lookout for parity separation will be caught off guard. Of course, you must adjust your patter at the end, as you express surprise that while neither colour nor odd/even separation was pulled off this time, curiously enough, the values of the cards in one pile are all prime, whereas the rest are all composite.
This month's entertainment is obviously inspired by Paul Curry's legendary "Out Of This World" from 1942.
"Terse tech" is an anagram of "the secret."
Colm Mulcahy (firstname.lastname@example.org) completed his PhD at Cornell in 1985, under Alex F.T.W. Rosenberg. He has been in the department of mathematics at Spelman College since 1988, and writing Card Colms---the only MAA columns to actively encourage lying on a regular basis---bi-monthly since October 2004. For more on mathematical card tricks, including a guide to topics explored in previous Card Colms, seehttp://www.spelman.edu/~colm/cards.html.
Follow Colm on Twitter at @CardColm