MATHEMATICAL MAGIC:
Finding The Card

by Chris Budd, University of Bath
mirrowed from Chris Budd's page, see Chris Budd's book [1] 


The mathematics behind this trick is quite deep. It uses an idea called a contraction map and a form of contraction mapping theorem which says that if you repeatedly apply a contraction map then you move towards a fixed point. This sounds rather dry but it leads to a spectacular card trick. The idea is that you perform a series of operations to a pack of cards which move the mystery card to a fixed point, which you know in advance. This tells you the location of the mystery card and you can use this to your advantage. I’ve used this card trick to motivate teaching of the contraction mapping theorem to third year undergraduates. This is much too advanced for Key Stage 3, but the trick is great in its own right, and the explanation of why it works is easy enough to understand (it is really a repeated application of division by 3).

The trick works by dealing 21 cards into three columns of seven cards. A volunteer selects a mystery card and says which column the card is in. This column is placed between the other two columns and the cards are dealt again. The process of selecting a column and placing it in the middle is repeated two more times. At this point the mystery card is at the middle of the pack, card number 11.

Suppose that x is the position of the mystery card in the pack (so that x lies between 1 and 21). This card will lie in one of the three columns and will be y cards down from the top of the column. When a column of 7 is placed above this then in the new pack the position of the card is now 7 + y. Because the cards are dealt into three columns, it is not hard to show that y =[x/3] where [z] means “round up z to the nearest whole number !” Thus if x is the original position of the card, its new position is

7 + [x/3]

It is easy to see that

11= 7 + [11/3]

so that the eleventh card stays where it is. Take any other number and apply the operation

x → 7 + [x/3]

three times. You find that you always get to 11 and stay there. Thi is how the trick works.

References:

[1] C. J. Budd and C. J. Sangwin.
Mathematics Galore!: Masterclasses, Workshops, and Team Projects in Mathematics and Its Applications.
Oxford University Press, USA (July 12, 2001)
ISBN-10: 0198507690
A book preview is available at Google Books: