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Card Colm
(by Colm Mulcahy)
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April 2011
Twisting the Knight Away (No Big Deal)
Hand out a deck of cards, requesting that a Jack and about a dozen number cards
be extracted. Have the selected cards well mixed, and set the rest of the deck
aside. Take the chosen cards back and fan through them, face up, joking, "These
could be mixed more, I can't help noticing that they are all facing the same
way!"
Turn some cards over, and shuffle the quarter deck a little. Fan again, flipping
the packet over if necessary, until you spot the Jack. Say, "The goal is to make
this Knight vanish into the packet so that none of us knows where he is. There's
a twist: you get to make him disappear, by thumbing off between one and six
cards, and flipping them over like this, as many times as you like."
Suppose the Jack is face-up in position four from the top of the fan: thumb off
four, five or six cards as a unit, and then flip it over, replacing the cards on
top of the rest of the packet, now reversed. The Jack will no longer be visible
if the cards are fanned. Now turn the entire packet over and repeat: thumbing
off and flipping over the same number of cards as before. Explain that you think
of this thumb and flip routine as "twisting" and also that it must always be done
to each end of the packet, hence the turning over and repeating using the same
number of cards.
Continue, "It doesn't matter if you know where the Knight is, as long as I don't
have any idea. I want you to use this die here to randomize things. Roll it,
and whatever number comes up, twist that many cards, as explained before. Then
turn the whole packet over and twist the same number of cards again. Then roll
the die once more, and use that number to determine how many cards you twist,
twice as did just now. I'll watch the first time so that I know you've got the
hang of it, then I'll turn away and you can do it as often as you wish. When
you're convinced that the Knight is well and truly lost, fan the cards to make
sure he's not face-up, then hand the packet to me."
Turn away as promised, and have the die rolled and double twisting done quite a
few times. Finally, you turn back, and are handed the packet. Fanning it
quickly, you have no difficulty correctly identifying the Knight.
Low-Down Dealing Without the Dealing
The above is an impromptu version of a more interesting effect which we shortly
discuss in some detail. Essentially, it's based on a twist on the triple and
quadruple dealing of the first Card Colm (October 2004), in the related double dealing
incarnation studied here last time around (February 2011). For the sake of
completeness, we also consider the hitherto unmentioned Low-Down Single Dealing.
Before exploring the deal-free alternative, let's review the basic move. Start
with a packet of n cards, and pick a number k such that n/2
≤ k ≤ n. Have k cards dealt out into a pile,
thus reversing their order, holding the cards low down, and have the remaining
n - k cards dropped on top as a unit. This is the deal about this
DALHDROP (Deal At Least Half and Drop the Rest On Top) move:
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If this deal is performed just once, the "top half" of the packet and the
"bottom half" of the packet switch places, so that for instance if the packet
starts with eight red cards on top of eight black cards, it will end up with
with eight black cards on top of eight red cards, subject to some rearrangement
within each such half. If the packet has odd size, then the middle card is
fixed and the "halves" refer to the cards lying above and below this fulcrum.
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If this deal is performed twice, then cards equidistant from the center remain
equidistant from the center. This palindromic principle was explored here a
little in February 2011. Of course, if the deal is performed twice, then
the "top half" of the packet is also restored to the top.
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If this deal is performed three times, the bottom k cards become the top
k cards, in reverse order. The top and bottom halves are switched in a
very orderly way: not only only are the bottom k cards transformed into
the top k cards in reverse order, but the original top n-k cards
are also transformed into the bottom n-k cards in the same order. For
instance, applying DALHADTROP three times with nine cards from a packet of
twelve cards arranged Ace, 2,
3, ..., 6, 7, ..., Jack,
Queen, from top to bottom, yields Queen, Jack, ..., 7,
6, ..., 4, Ace, 2, 3.
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If this deal is performed four times, the packet is restored to its original
order.
As discussed in the first Card Colm (October 2004), the last two
claims above can be seen by studying the sequence of images here:
This illustrates the case n = 13 and k = 8, where the packet after
successive DALHDROP moves is represented by vertical stacks of grey panels,
from left to right, the panels initially being in decreasing order of brightness.
We now explain how to have a similar effect on a packet of cards without any
dealing and dropping at all. The key is the observation that:
The DALHDROP (Deal At Least Half and Drop the Rest On Top) move
essentially divides a packet into two subpackets, reverses the order
of the cards in the larger one, and reassembles into a single packet.
There are other ways of achieving the same goal, such as reversing the order of
the cards in the smaller subpacket!
Applying DALHDROP once to nine cards from twelve face-up cards arranged Ace, 2, 3,
..., 6, 7, ..., Jack, Queen, from top to bottom, yields
10, Jack, Queen, 9, 8, 7, 6, ...,
Ace.
What if, instead of doing that, one simply thumbed off the top nine cards as a
unit, flipped it over, and tucked it underneath the remaining three cards? The
only difference between the final outcome here and that earlier would be that
nine of the cards would now be facing the opposite way.
Here's another alternative with even less thumbing off: once again start with
a face-up arrangement Ace, 2,
3, ..., 6, 7, ..., Jack,
Queen, from top to bottom. Turn the whole packet over, so it's now face-down,
and then thumb off just three cards, as a unit, holding them to one side. Take
the nine cards underneath and bring them forward, turning them over as a unit,
and dropping them on top of the three thumbed off. Or one could thumb off three
cards, flip them over as a unit, and replace them on top of the remaining
nine--perhaps turning the whole packet over at the end! Ignoring card
orientation issues, either of these yields similar results. In what follows,
we'll apply such alternative moves to both ends of the packet, so we can
dispense with any initial or final turning over of the whole packet.
The video clips below illustrate in the case n = 13 and k = 8. We
use Clubs, in ascending order. The first video clip shows the standard DALHDROP
move applied to face-down cards. The second shows the new move, applied to the
same cards, where five cards are thumbed off instead of eight being dealt out.
The third clip shows how one can make it less obvious what is really going on:
by starting with the cards in some desirable order, but having some face-down and
others face-up.
Click on the images below to see the clips.
The key here is to: thumb off and flip at most half of the cards, maybe
we should call this the TOFAMH (Thumb Off and Flip At Most Half) move.
Now we are ready to try our first application. Start with all thirteen Clubs,
where Ace = 1, Jack = 11, etc. Pair them as follows: 2 with 7, then 3 with 9,
then 4 with 10, and 5 with 8, and finally 6 with the Jack. Note that each number
is paired with its multiplicative inverse, modulo 13. Pair the Ace and Queen too.
Arrange these twelve cards in a "palindromic stack," with the King in the middle.
The elements of each pair as just assigned are equidistant from the center. Turn
over one card in each pair so that it faces the other way. Hence, the 6 is as
far from the center card as the Jack is, and is facing the other way.
No matter how much "double twisting" is done, as explained earlier, the Jack and
6 will remain equidistant from the King, and facing opposite ways, so that all
you need to do upon reclaiming the cards is look for the face-up 6, and the Jack
(or Knight) will be in the corresponding position, either counted out from the
middle or in from the end.
Furthermore, every face-down card---and there will be six or seven overall---can
also be speedily identified. This suggests options such as having people pick
two or three of those, instead of playing the game of Twisting the Knight Away.
There is no particular significance to the multiplicative inverse pairings above,
it just has the advantage of being both easy to remember and hard for the average
audience to spot any pattern.
For the impromptu version we started with, you just need to note the card which
is the same distance from the center as the Jack at the outset. Of course it
must be facing the opposite way; another quick shuffle and a glance at both sides
of the resulting fan should do the trick.
Colm Mulcahy (colm@spelman.edu)
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, see
http://www5.spelman.edu/~colm/cards.html.
Video credit: Molly Mulcahy.
Follow Colm on Twitter at @CardColm
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