A Thousand Dollars And a Bunch of Envelopes

Mar 06, 2006

RAY: You know what inspired me to use this puzzler? All of these envelopes that are in front of us. We have dozens of envelopes sitting on the table.

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, and how much money do you put in each one?
RAY: In the first nine envelopes, you're going to put one dollar, 2 dollars, then 4, 8, 16, 32, 64,128, and 256. That's nine envelopes.

If you add all those up, that's 511 dollars.

So if I ask you for any amount up to 511 dollars, you would give me some combination of those envelopes, and if I asked you for 511

TOM: I'd give you all of them.

RAY: Right. So, what do I put in the next envelope?

TOM: Everything else.

RAY: Exactly. 489 bucks. And if you do that, you can give me any amount of money that I ask for. For example, if I ask for say, 671 dollars, you would give me the envelope that's 489 dollars, plus you'd give me another 182 dollars. But we know that you can give me 182 dollars.

TOM: You could do that combination with the other envelopes. I like it.

RAY: Cute, eh? So who's our winner?

TOM: The winner is Linda Williams from Ann Arbor, Michigan. And for having her answer selected at random from among all the correct answers that we got, Linda gets a 26-dollar gift certificate to the Shameless Commerce Division at Car Talk.com with which she can get one of our fine -- you know when something is really junk, that's when you use the adjective "fine" to describe it -- anyway Linda, you can get one of our fine 'Best of Car Talk' collections on CD.

That's a fine product isn't it?

RAY: Oh yes. The finest!

Get the Car Talk Newsletter