Riddle: So you have a million dollars

Miscellaneous and Personal Stuff I Must Share
1

In one-dollar bills. You also have ten envelopes.

Is it possible to put a certain number of dollar bills in each envelope such that–once the relatives come a-knockin’–that regardless of the amount they request, you can hand over a certain combination of the envelopes, and give them exactly that amount?

I heard this riddle from a co-worker. I can see how it could be done with $1000, but not a million.

2

I’m pretty sure it’s impossible… but here’s an old, but possible puzzle (slightly adapted to make it not too off topic)…

Can you put the bills into the envelopes in such a way that half the envelopes contain an odd number of bills and the remainder contain an even number of bills?

(If the orignal problem does have a solution, my apologies in advance for my conviction otherwise)

3

Nope, but you can do it with 20 envelopes.

envelope 1 = $1
envelope 2 = $2
envelope 3 = $4
envelope 4 = $8
envelope 5 = $16
envelope 6 = $32
etc.
envelope 20 = $544448

I can’t think of a way to do it with 10.

Unless you don’t seal the envelopes, and then when they ask for their money, you re-arrange things. :slight_smile:

4

I’m pretty sure it’s impossible too…

It did remind me of the one: how many weights required to weigh any whole number of kilos up to 40?

But I don’t see how the same principle could be applied in this case…maybe the question’s been corrupted along the way? Or more likely, it’s a trick question…
Oh, if you haven’t heard it before (and can’t work it out :slight_smile: ), the answer for the weights is 4.

5

Yeah, here’s proof it’s impossible. Label the envelopes 1 through 10.

You either give them envelope 1, or you don’t. (2 choices)

You either give them envelope 2, or you don’t. (2 choices)

You either give them envelope 3, or you don’t. (2 choices)
.
.
.
.
You either give them envelope 10, or you don’t. (2 choices)

Count the number of different combinations of envelopes you can give–it’s 2[sup]10[/sup] = 1024. So there can be at most 1024 different values of money you can give, which is certainly less than the 1,000,001 different possible amounts that can be asked for.