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### 1000 \$1 Bills in 10 Envelopes

```Date: 12/11/2002 at 04:10:54
From: Kendel
Subject: Word problem

You are given 1000 one dollar bills and 10 envelopes. Put the bills
into the envelopes in such a way that someone can ask you for any
amount of money from \$1 to \$1000 (examples - \$532, \$619, \$88, etc.)
and you can give it to them through a combination of the envelopes.

I am NOT a math person by any means and I'm sure that the answer to
this problem is painfully obvious; however, I can't get it! I can do
it with 12 envelopes like this - 400, 300, 100, 100, 40, 30, 20, 10,
5, 2, 2, 1, but I can't get it down to ten. I'd appreciate a response
as soon as you have time.

Thank you,
Kendel
```

```
Date: 12/11/2002 at 10:19:20
From: Doctor Ian
Subject: Re: Word problem

Hi Kendel,

If you can add 23 dollars of your own, then it's easy, since you can
then use each envelope to represent one place value in a binary
expansion:

512 256 128 64 32 16 8 4 2 1

Now you can get any number between 1 and 1023, e.g.,

9   8   7   6   5   4   3   2   1   0
2   2   2   2   2   2   2   2   2   2

512 256 128  64  32  16   8   4   2   1

\$1 =                                       x

\$5 =                               x       x

\$143 =           x               x   x   x   x

\$1023 =   x   x   x   x   x   x   x   x   x   x

But if you only have \$1000, you can only get 489 into the final
envelope.  Does this make a difference?

Well, you can still use the first 9 envelopes to return any amount up
to \$511.  So with the final envelope you can get anything between

\$489 + \$1 = \$490

and

\$489 + 511 = \$1000

So it works fine.

Does this make sense?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Place Value
Elementary Puzzles
High School Number Theory
High School Puzzles