What’s the maximum number of 3-sized subsets of $[n]$ that can exist such that no two subsets contain more than one common element? When $n \equiv 1,3 \mod 6$ then this is equivalent to a Steiner triple system. Each number will appear $\displaystyle\frac{n-1}{2}$ times and there are $\displaystyle\frac{n(n-1)}{6}$ subsets. But what about when $n \not\equiv […]

I am doing research on the sense of smell and have a combinatorics problem: I have 128 different odors (n) and I mix them in mixtures of 10 (r). There are 2.26846154e+14 different mixtures. What I want to know is: What is the size of the group of mixtures such that no mixture in the […]

A friend of mine asked me this question. “I have $3n$ elements, and I want to know which is the maximum number of triplets $(a,b,c)$ so that no two triplets have more than one element in common”. The first thing which came into my mind was the Kirkman’s schoolgirl problem: with 15 elements there are […]

Given a solution to Kirkman’s School Girl Problem, it is of course easy enough to check that it actually is a solution. But how could you reconstruct it if you lost it? Is there a method or algorithm for constructing a solution which is easier to remember than the actual solution? There are many combinatorial […]

How many different ways can the signs be chosen so that $\pm 1\pm 2\pm 3 … \pm (n-1) \pm n = n+1$? This is an extension of this question: For what $n$ can $\pm 1\pm 2\pm 3 … \pm (n-1) \pm n = n+1$? There I asked “for what values of $n$ can the signs […]

I am coordinating cookie-baking events with kindergarten kids. This turns out to be a challenging problem, and I could use a little help: We would like a general way of creating ‘fair’ cookie-baking teams for kindergarten pupils. By ‘fair’ I mean the following: A class consisting of N children (usually in the range 18 to […]

I’m trying to build a 3D matrix made up of positive integers that has very specific properties. The matrix dimensions are $N \times N \times (N+1)$ where $N$ is a positive integer. The matrix has two properties: Every one of the $(N+1)$ “slices” of size $N \times N$ of the matrix contains each of the […]

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