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The trick for N = 10 (which I heard from a friend earlier today) is to check that the density of the triangular packing of unit diameter circles is high enough that some translate of this packing must …

Is there a discrete analogue of the notion of discreteness?

Yes, it has been studied by Johan Wästlund in A solution of two-person single-suit whist, which gives an efficient algorithm to compute the value of a position in this game (Theorem 10.1). He has als …

If my computation is correct, then f(n, 2) should be roughly $$\frac12 \sum_{k \in \mathbb{Z}} 2^{k+s} e^{-2^{k+s}}$$ where s = the fractional part of $\log_2 n$. (Note the terms of the sum decay r …

A permutation of {1, ..., n} with k fixed points is determined by choosing which k elements of {1, ..., n} it fixes and choosing a derangement of the remaining n-k elements. So, $F(k, n) = {n \choos …

The case k = 1 is the Erdős-Ginzburg-Ziv Theorem. Take a look at this Wikipedia article which has links to some surveys of the large literature of similar results. (The particular generalizations I' …

This is just a guess with no basis, but maybe $D_n^+(q)$ should be those elements whose determinant is a quadratic residue (maybe let's assume p > 2 for safety)? Or you could split into $q-1$ groups, …

This problem seems to be NP-hard, in an informal sense. I'll sketch how we could use an algorithm for this problem to solve the knapsack problem. Suppose given $n$ objects with weights $w_1$, ..., $ …

You might want topos theory. A topos is something like the category of sets, but the internal logic of a general topos is much weaker than ZF; it need not even be Boolean. An example of a topos is t …

The method of coupling from the past can be used to sample uniformly at random from certain distributions. Here is a simple demonstration which illustrates the method, and here it is in action comput …

Writing B \ A for the event "B occurs but A does not" (as in the difference of sets) we have... P(A ∪ B) = P(A) + P(B \ A) P(A ∩ B) = P(A) × P(B | A) Just fun with symbols I think...

Why isn't there an intrinsic topological description, or perhaps manifold-theoretic description? At least in some cases, the combinatorial Euler characteristic of X is equal to the homotopy Euler …

I'm very curious where this problem comes from, since it is related to some stuff I've been thinking about. The smallest counterexample for k=1 seems to be the set of six points containing the vertic …