# 1000 \$1 Bills and 10 Envelopes

##### May 06, 2017

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 configure those 10 envelopes, that is, 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: I gave Tommy a thousand \$1 bills, and his assignment was, first, come up with 10 envelopes, and once he did that he had to figure out a way to put various numbers of dollar bills in those 10 envelopes so that no matter what amount of money I asked him for, he could hand me some combination of envelopes and always be sure of giving me the exact right amount. The question is: How did he do it?

TOM: That's the question? I thought the question was, “What was going on in your mind that you gave me 1,000 \$1 bills?”

RAY: I could've given a hint last week and said one of the envelopes has \$489 in it.

TOM: Really?

RAY: Yes. And 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…

TOM: 512.

RAY: 512, OK? Two to the tenth would be 512, but he couldn't put 512 because you don't have it in there.

TOM: No.

RAY: 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.

TOM: Man.

RAY: Pretty good, huh? Give me my thousand bucks back. It was only a loan.

TOM: Give me my envelopes back.

RAY: A further demonstration of the power...

TOM: Of two.

RAY: You can't do it unless the number is two. There you go, see? The power of two. Right?