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 
easy to use in your head.

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/