1000 $1s in 10 Envelopes

Jan 19, 2013

RAY: I'm going to give you a thousand $1 bills. You come up with 10 envelopes.

Here's your assignment: Figure out a way to put various numbers of dollar bills in those 10 envelopes, so that no matter what amount of money I ask you for, you can hand me some combination of envelopes and always be assured of giving me the correct amount of cash.

TOM: Let me get this straight. If you say, "Give me $637," I can say, "Oh, that will be envelope number one, envelope number six, and envelope number two."

RAY: You got it.
RAY: All right. Here’s how you solve it.

I could have given a hint last week and said one of the envelopes has $489 in it. And the other envelopes...the other nine have $1... $2... $4... $8, $16, $32, $64, $128, and the ninth envelope has $256. If you add those up -- 256, 128, 64, 32 -- you come up with 511, because in base 2, the next number would be... 512.

Two to the tenth would be 512, but he couldn't put 512 because you don't have it in there. So you could put 489. So you can get any possible number between one and 511 by using the first nine envelopes, and then anything beyond 511 up to a 1,000 using 489 plus one gives you 490, 490 plus two gives you, and da-da-da-da-da.

A further demonstration of the power of two. You can't do it unless the number is two. Do we have a winner this week?

TOM: We do. The winner is Betty Benson from Houston, Texas. Congratulations, Betty! 

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