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Cardiograph
`Cardiograph' is a brand new card effect developed by the writer and is
presented here in print for the first time. Since the method of
operation is rather unusual and because the basic principle is so deeply
hidden, it is doubtful whether even a competent mathematician could
solve its method of working.
Presentation
The performer hands out a deck of cards to a spectator with the request
that after shuffling the cards, he is to secretly remove a packet of
from one to fifteen cards. Taking back the depleted deck, the performer
states that he in turn will also cut off a small packet of cards. The
spectator is then instructed to inspect his cards so that he may be
completely satisfied that they represent a random selection. The
performer also inspects his packet of cards for the same reason.
The spectator is now requested to place his packet of cards face down on
the table following which the performer places his face down packet on
top of it. At this point, neither party has any knowledge of the total
number of cards assembled in the pile on the table nor is this necessary
information. The performer now explains that one card in particular will
be selected from this pile by a straightforward dealing routine. The
spectator is asked to pick up the cards and deal them alternately face
down into two piles, the first card being dealt to the performer, the
second to the spectator, the third to the performer, the fourth to the
spectator and so on until the packet is exhausted. Of the two dealt
piles, the performer's pile is discarded. The pile in front of the
spectator is now picked up by the spectator and a second alternating
deal out is made as before (first card to performer). At the end of the
second deal the performer's pile is again discarded. This dealing
routine is continued until only one card is left in the possession of
the spectator ( a one card pile). Note that the alternating deal must
start with the performer each time and that at the end of the deal it is
always the performer's pile of cards that is to be discarded.
The one card left with the spectator is now placed face down on a piece
of colored paper ( or aluminum foil) that the performer explains has
been treated to emit gamma rays that penetrate only the printed portion
of the playing card. As a result, the face of the card is outlined in
reverse on the back of the card and in effect provides a cardiograph of
the card when viewed from the back. However, a certain amount of
know-how and concentration is necessary to detect the shadowy outline
that is given by the gamma ray penetration.
Method
When the performer is handed back the deck he secretly counts off and
removes exactly eight cards. Under the pretense of inspecting them for
randomness of distribution he actually memorises the bottom and the
third from the bottom cards of his face down packet. Hence, when he
places his packet of cards face down on the spectator's face down packet
on the table he knows the value of these cards.
They correspond to the sixth and eighth card positions counting down
from the top of the assembled packet on the table.
On the first deal out, the performer makes sure that he is dealt the
first card ( this would be a standard way of dealing out cards anyway by
the spectator). On the second time around, however, not only does he
make sure that he is dealt the first card but he must also note whether
he or the spectator gets the last card. If the performer gets the last
card at the end of the second deal the spectator will finally end up
with the card that the performer spotted at the bottom of his eight card
packet at the start of the trick. Conversely, if the spectator gets the
last card at the end of the second deal, the spectator will end up with
the card that the performer spotted as third from the bottom of his
eight card packet. The rest of the trick is self working. The performer
knows the one card that will end up in front of the spectator ( he
forgets the other card that no longer applies to the situation) and it
only remains for the spectator to complete the remaining deal-outs and
the performer to introduce whatever dramatics are necessary to properly
impress the audience with his cardio graphic vision.
Explanation
This original card effect is based on the statistical observation that
when a packet of from eight to twenty-three cards is dealt out in the
manner prescribed under the presentation, either the sixth or eighth
card ( counting down from the top of the face down packet) will end up
as the last remaining card in front of the spectator. Table 2-1 lists
the actual data.
TABLE 2.1
Number Of
Cards In Initial
Packet
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Card From Top of Pocket that will End Up in Front of Spectators
6th 8th 6th 8th 6th 8th 6th 8th
Note that if the total number of cards were known, the performer could
tell at this point whether the sixth or the eighth card from the top
would end up in front of the spectator ( corresponding to cards he
memorised earlier ). Unfortunately the performer does not know the total
number of cards in the assembled packet on the table nor is an estimate
good enough. However, resort can be made to the following mathematical
expedient to establish whether or not the 6th or 8th card is destined to
be the last card to end up in front of the spectator.
Observe that in Table 2-1 there is a systematic alternation between the
6th and 8th positions.
Consider first only initial card packets containing an even number of
cards. On the first alternating deal, such cards are distributed into
two piles, each having an equal number of cards. In turn these piles
will contain cards that are either odd or even in number. By reference
to Table 2-1, it is seen that the initial packets containing 8, 12, 16,
and 20 cards are associated with the card in the sixth position. But
note that these are also the packets that give even numbered piles when
dealt into two heaps ( 4, 6, 8, and 10 cards per heap respectively ). In
the same way, it is also seen that the initial packets containing 10,
14, 18, and 22 cards are associated with the card at the eighth
position. But these are exactly the packets that give odd numbered piles
when dealt into two heaps ( 5, 7, 9, and 11 cards per heap
respectively).
So evenness and the sixth position and oddness and the eighth position
become bracketed together. This fact of evenness or oddness is reflected
in the second deal. If the performer gets the last card on the second
deal, the spectator's packet must have contained an odd number of cards.
In turn the performer associates this with the card at the eighth
position (the card he memorised at the bottom of his original eight card
packet ). If the spectator gets the last card on the second deal, the
spectator's packet must have contained an even number of cards. In turn
the performer associates this with the card at the sixth position ( the
card he memorised as third from the bottom of his original eight card
packet). Hence by knowing who receives the last card on the second deal
the performer knows which of his two memorised cards he must retain in
his memory and disclose at the end of the trick.
The foregoing argument applies only to a consideration of those initial
card packets containing an even number of cards. However, it will now be
shown that this argument is equally applicable to initial card packets
containing an odd number of cards.
On the first alternating deal of a card packet containing an odd number
of cards, cards are distributed into two piles, but this time the two
piles will be unequal in number (the performer's pile will have an extra
card ). This means, in turn, that the spectator has the same number of
cards that he would have received had the odd-numbered packet contained
one less card. Thus whether an initial packet for dealing had say 8 or 9
cards, 16 or 17, or 20 or 21 cards, the spectator receives only 4, 8, or
10 cards respectively for the cards in his pile at the end of the first
dealing. Hence in all cases, the same card positioning that applied to
the second deal for initial packets with an even number of cards applies
equally well for initial packets with an odd number of cards.
Comment
By restricting the number of cards that are to be removed by the
spectator to fifteen, the total number of cards that can possibly be
dealt out ( including the performer's packet of eight cards) will range
between nine ( 9=1+8) and twenty-three (23=15+8 ). This brings the
numbers involved in this card effect into the region where the
uniqueness of the 6th and 8th card positioning is valid.
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