Henry Ernest Dudeney/Modern Puzzles/11 - The Perplexed Banker

by : $11$

 * The Perplexed Banker
 * A man went into a bank with $1000$ sovereigns and $10$ bags.
 * He said,
 * "Place this money, please, in the bags in such a way that if I call and ask for a certain number of sovereigns
 * you can hand me over one or more bags, giving me the exact amount called for without opening any of the bags."
 * How was it to be done?


 * We are, of course, only concerned with a single application,
 * but he may ask for any exact number of pounds from $\pounds 1$ to $\pounds 1000$.

Solution
For $n = 1$ to $9$, put $2^{n - 1}$ sovereigns in bag $n$.

Put the remaining $489$ in bag $10$.

Proof
A thinly-disguised implementation of the Basis Representation Theorem for binary.