Boxes of Money

Question: You are given b boxes and n dollar bills. The money has to be sealed in the b boxes in a way such that without thereafter opening a box, you can give someone a requested whole amount of dollars from 0 to n. How should b be related to n for this to happen?  Answer: Stumped? Let’s think of an example to approach this problem.  Say we have $100. A good approach to distributing $100 would be the binary number system. So you’d have $1, $2, $4, $8, $16, $32 in the first six boxes. We can’t fill the next box with $64 dollars because we are only left with $37 dollars (from a total of $100). So we’d have to put $37 in the seventh box. To supply any requested amount, we’d have to use a combination of these boxes.  To find out the restrictions on the values of b and n, we have to think of different scenarios. For instance, with a million dollars and just one box, we would never be able to dispense any requested amount less than a million. However, if we are ever in a situation with more boxes than dollars, there is a never a problem.  Using this approach, we can create a table showing the best relationship between b and n  b = 1     n = up to $1 b = 2     n = up to $2 + $1 = $3 b = 3     n = up to $4 + $2 + $1 = $7 b = 4     n = up to $8 + $4 + $2 + $1 = $15