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### Six-Card Trick

```
Date: 10/25/2000 at 05:39:06
From: Richard Marsh
Subject: Logic problem

I can't figure this one out.

There are three participants: a Host, a Partner, and a Volunteer. The
Partner is in a soundproof room.

The Host gives the Volunteer six blank cards, five white and one blue.
The Volunteer writes a different integer from 1 to 125 on each card as
the Host watches. The Volunteer keeps the blue card and gives the
others to the Host.

The Host arranges the five white cards in some order and passes them
to the Partner. The Partner then announces the number on the blue
card. How?

No tricks with turning cards around to give extra information, just 5
numbers in a certain order.

Thanks,
Richard
```

```
Date: 10/25/2000 at 10:55:00
From: Doctor Rick
Subject: Re: Logic problem

Hi, Richard.

Let's see if I understand the problem. The Volunteer isn't in on the
stunt, right? He just picks six random numbers; he may even try to
make it harder for the Host and Partner. The Host uses some means to
signal the number to the Partner, who knows the code the Host is
using. Is this correct?

The numbers on the white cards are all different, so whatever they
are, it's equivalent to putting the numbers 1 to 5 in some order (just
call the third lowest numbered card "3", etc.) How many ways can the
cards be ordered? There are 5! = 5*4*3*2 = 120 permutations.

If you don't know what I mean, see "Permutations and Combinations" in
our Dr. Math FAQ:

http://mathforum.org/dr.math/faq/faq.comb.perm.html

That's just enough, because the Partner knows not only that
the number is between 1 and 125, but that it is not one of the five
numbers he is given.

I'll leave it to you to come up with a particular code if you're
interested. What I've said is enough to prove that the trick can be
done in principle. It might be a challenge to develop a code that's

You've also raised an interesting question: how many numbers could be
coded if he communicates additional information by turning some cards
around? This doesn't sound so hard, if there is a definite top and
bottom to each card. However, if the cards are completely blank to
start, and the Volunteer chooses numbers like 81, 96, etc. (numbers
that could be read from either side), the Partner won't know whether
they are right side up or upside down. In this case, the Volunteer
might be able to thwart the communication by choosing such numbers.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
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