000RM.dvi

336 Cheney’s card trick

12.1 Principles

You are the magician’s assistant. What he will do is to ask a spectator to
give you any 5 cards from a deck of 52. You show him 4 of the cards,
and in no time, he will tell everybody what the 5th card is. This of course
depends on you, and you have to do things properly by making use of
the following three basic principles.

1.The pigeonhole principle. Among 5 cards at least 2 must be of the

same suit. So you and the magician agree that the secret card has

the same suit as the first card.

2.The distance of two points on a 13-hour clock is no more than 6.

We decide which of the two cards to be shown as the first, and

which to be kept secret. For calculations, we treatA,J,Q, andK

are respectively 1, 11, 12, and 13 respectively.

Now you can determine the distance between these two cards. From

one of these, goingclockwise, you get to the other by travelling this

distance on the 13-hour clock. Keep the latter as the secret card.

Here are some examples.

hours distance clockwise

2 and 7 5 2 to 7

3 and 10 6 10 to 3

2 andJ 4 Jto 2

Aand 8 6 8 toA

3.There are 6 arrangements of three objects.

The remaining three cards can be ordered assmall,medium, and

large.^1 Now rearrange them properly to tell the magician what

number he should add (clockwise) to the first card to get the number

on the secret card. Let’s agree on this:

arrangement distance

sml 1

slm 2

msl 3

mls 4

lsm 5

lms 6

(^1) First by numerical order; for cards with the same number, order by suits:♣<♦<♥<♠.