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The purpose of this answer is to use the second moment method to make rigorous the heuristic argument of Michael Lugo. (Here is why his argument is only heuristic: If $N$ is a nonnegative integer ran …

This is an answer to your "actual question" (2), building on some of the ideas in Douglas Zare's answer. Lemma 1: Suppose that $0 < r < 1$. Let $S=\lbrace \epsilon r^i : \epsilon = \pm 1 \text{ and …

For any $t$, if $m$ is sufficiently large relative to $t$, and $n$ is any positive integer (possibly equal to $m$), then the circle method proves that there exists an infinite sequence of increasingly …

Here is a sketch of a solution; the details would be tedious, but doable by someone with enough patience I think. The point is that the terms with $i$ and $j$ both of the form $d/2 + O(\sqrt{d})$ are …

What is the length $f(n)$ of the shortest nontrivial group word $w_n$ in $x_1,\ldots,x_n$ that collapses to $1$ when we substitute $x_i=1$ for any $i$? For example, $f(2)=4$, with the commutator …

Your problem is hard, but here are some things that can actually be proved! Let $S_d$ be the set of polynomials of degree $d$ with all $d+1$ coefficients in $\{\pm 1\}$. 1) $|S_d| \gg 2^d/d$ as $d \ …

Any general algorithm for this problem will require exponential time. In fact, just writing down the answer can take exponential time in some cases. For example, suppose that $n=2m$ for some odd $m$ …

(This is a variant of Cartier's argument mentioned by Dan Ramras.) Let $X$ be a finite set of size at least $2$. Let $E$ be the set of edges of the complete graph on $X$. The set $D$ of ways of dire …

Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication "If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true." for ev …