Smallest Set of Weights for One-Pan Balance

Classic Problem
Consider a balance for determining the weight of a physical object.

Let this balance be such that weights may be placed in one of the pans.

What is the smallest set of weights needed to weigh any given integer weight up to a given amount?

Solution
A set of $m$ weights in the sequence $\sequence {2^n}$:
 * $1, 2, 4, 8, 16, \ldots$

allows one to weigh any given integer weight up to $2^m - 1$.

Proof
This is equivalent to the statement that every positive integer can be expressed uniquely in binary notation.

This in turn is an application of the Basis Representation Theorem.

Also see

 * Smallest Set of Weights for Two-Pan Balance