Postage stamp problem

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The postage stamp problem (also called the Frobenius Coin Problem and the Chicken McNugget Theorem ^[1]) 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.^[2]

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.

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.

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.^[2]

• Coin problem
• Knapsack problem
• Subset sum problem

• Lunnon, W. F. (1969). "A postage stamp problem". Comput. J. 12 (4): 377–380. doi:10.1093/comjnl/12.4.377.
• Alter, R.; Barnett, J. A. (1980). "A postage stamp problem". Amer. Math. Monthly. 87 (3): 206–210. doi:10.2307/2321610. JSTOR 2321610.
• Graham, R. L.; Sloane, N. J. A. (1980). "On additive bases and harmonious graphs". SIAM J. Algebr. Discrete Methods. 1 (4): 382–404. CiteSeerX 10.1.1.70.5521. doi:10.1137/0601045.
• Challis, M. F. (1993). "Two new techniques for computing extremal h-bases A_k". Comput. J. 36 (2): 117–126. doi:10.1093/comjnl/36.2.117.
• Kohonen, J.; Corander, J. (2013). "Addition chains meet postage stamps: reducing the number of multiplications". arXiv:1310.7090 [math.NT].
• Kohonen, Jukka (2014). "A meet-in-the-middle algorithm for finding extremal restricted additive 2-bases". arXiv:1403.5945 [math.NT].
• Weisstein, Eric W. "Postage Stamp Problem". MathWorld.
• OEIS sequence A001212 (Solution to the postage stamp problem with n denominations and 2 stamps)