Card Tricks > Contest Card Tricks
Gamesmanship
Presentation
This mathematical card effect is actually a game or contest between
the performer and a member of the audience. The ground rules are as
follows. A number of cards, preferably fifteen or more, are spread out
on a table to be designated by the spectator who will oppose the
performer. Cards are removed alternately by the performer and the
spectator. The winner of the contest is the one who can force his
opponent to take the last card under the stipulated rules.
Method
If the performer makes the first drawing, he can generally force his
opponent to take the last card by resorting to the following
mathematical expedient. The performer works with two numbers, the total
number of cards that are initially spread out on the table for
withdrawal and the maximum number of cards that can be withdrawn at any
turn. As a first step, he subtracts the number of one from the total
number of cards and then divides his number by a number that is one more
than the maximum number of cards than can be withdrawn at any turn. The
remainder left over from this division is the number of cards that he
first picks up. He then notes the number of cards picked up by his
opponent and each time on his turn the performer removes a number of
cards removed by both his opponent and himself equal to the number of
cards that can be removed at any one time plus one.
An example for the computation will clarify the process. Assume that the
spectator calls for 23 cards to spread out on the table and that no more
than 5 cards can be removed at a time. The performer subtracts 1 from 23
giving 22 (22=23-1). The add 1 to 5 making (6=5+1). He continues by
dividing 6 into 22 giving a remainder of (4=22-36). The performer
removes four cards as the game starts and he notes how many cards his
opponent removes and adjust the number of cards he withdraws so that the
total withdrawn by his opponent and himself total 6 for their combined
turns.
The performer will always win provided a remainder is left over when the
maximum number of cards that can be withdrawn plus one is divided into
the total number of cards initially spread out on the table minus one.
TABLE 6-1
Four Cards Are First Withdrawn By the performer After Which the
following Schedule Applies:
Cards withdrawn by spectator Cards then withdrawn by performers
Total cards removed for combined turn
1 2 3
4 5
5 4 3
3 1
6 6 6
6 6
However, this is not necessarily so when no remainder is left over.
He can persuade his opponent to make the first withdrawal or the
performer can start by taking one card and hope that his opponent will
take less than the required number plus one. Whenever this occurs, the
performer seizes this opportunity to complete the total with his next
withdrawal. He then proceeds according to the standard routine.
Explanation
Initially spread the cards out on the table total t and let the
maximum number of cards that can be removed at any turn equal m. The
point of the game is to force the other person to withdraw the last
card. Up to the stage of the last withdrawal, t-1 cards will have been
removed. To win, the performer anticipates this situation ahead of time
as follows. For any one round the performer secretly controls the number
of cards that are removed by both of them. Thus when his opponent
removes cards in number running from m to 1 respectively so that between
them they remove a total of m+1 cards for any one round. By dividing t-1
by m+1, the performer determines the number he must start with to
eventually arrive at t-1 with his opponent's turn to withdraw a card.
This number is the remainder r left over as a result of the division of
t-1 by m+1.
Card Tricks > Contest Card Tricks
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