Math Is Fun Forum

Math Guy

Member

Registered: 2005-10-12

Posts: 20

A Thousand Dollars—And a Bunch of Envelopes

Here’s the puzzler. It’s very simple.

I’m going to hand you one thousand dollars, in one-dollar bills.

Your job is to put some of those dollar bills in the envelopes, in such a manner that no matter what number of dollars I ask you for you’ll hand me the appropriate combination of envelopes.

TOM: There must be more to it, because I could just use a thousand envelopes.

RAY: There is more. The question is--what’s the fewest number of envelopes I can use, and how much money do you put in each one?

*Leave what you think is the right answer here, guessing never hurts!!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: A Thousand Dollars—And a Bunch of Envelopes

1, 2, 4, 8, 16, 32, 64, 128, 256, 489

10 envelopes

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,385

Re: A Thousand Dollars—And a Bunch of Envelopes

Excellent, Ricky!
I remember this question was asked to me about 15 years ago. I managed to say 1,2,4,8,16..etc. But 489 was a little difficult. I did that too! The most satisfying part of it was that I did it without a paper and a pencil. Ever since, this has been one of my favorite questions.
I was able to do it because I was too familiar with the powers of 2, and their sums,  not because of any other reason. (Modest me! )

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: A Thousand Dollars—And a Bunch of Envelopes

Same here.  I figured that you could either give or not give an envelope, and thus, the whole thing is represented as a binary system.  So it just makes sense to do it in base 2.

Being a computer scientist helped a bit too. 

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."