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/
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