Postage stamp problem

The postage stamp problem (also called the Frobenius coin problem and the Chicken McNugget theorem) is a mathematical riddle that asks what is the smallest postage value which cannot be placed on an envelope, if the latter can hold only a limited number of stamps, and these may only have certain specified face values.^[1]

13 is the smallest total that cannot fit on an envelope with space for only three stamps of possible values 1, 2, 5 and 20

For example, suppose the envelope can hold only three stamps, and the available stamp values are 1 cent, 2 cents, 5 cents, and 20 cents. Then the solution is 13 cents; since any smaller value can be obtained with at most three stamps (e.g. 4 = 2 + 2, 8 = 5 + 2 + 1, etc.), but to get 13 cents one must use at least four stamps.

Mathematical definition

Mathematically, the problem can be formulated as follows:

Given an integer m and a set V of positive integers, find the smallest integer z that cannot be written as the sum v₁ + v₂ + ··· + v_k of some number k ≤ m of (not necessarily distinct) elements of V.

Complexity

This problem can be solved by brute force search or backtracking with maximum time proportional to |V |^m, where |V | is the number of distinct stamp values allowed. Therefore, if the capacity of the envelope m is fixed, it is a polynomial time problem. If the capacity m is arbitrary, the problem is known to be NP-hard.^[1]

See also

• Coin problem
• Knapsack problem
• Subset sum problem