turkmath.org

If McNuggets come in boxes of 6, 9 and 20: (i) What's the largest number of McNuggets we can not buy? (ii) How many such numbers are there? (iii) How many ways are there of buying exactly n McNuggets? The questions above are asking for the Frobenius number, the genus and a computable answer to the enumeration problem for the numerical monoid generated by 6, 9 and 20. A numerical monoid is a co-finite subset of natural numbers closed under addition and containing 0. Numerical monoids are usually given by specifying a set of generators (which are positive integers, the additive monoid they generate is co-finite if and only if they are coprime). When there are only two generators a and b all three questions have definitive answers; for the first two they are (i) ab - a - b and (ii) (a-1)(b-1)/2. For three generators the first two questions have a satisfactory answer, but there is no effective closed formula known for the third (however there are polynomial time algorithms for any number of generators). For more than three generators the subject is wide open. While some of these questions can be explained to schoolchildren, the subject is intimately connected to algebraic and convex geometry, combinatorics, commutative algebra, topology, dynamical systems, etc. I'll try to give a taste of some of these connections, sketch new elementary proofs of some classical results and mention some recent developments.