The recent breakthrough paper of Justin Gilmer on the Union-Closed Set conjecture contains this mysterious coincidence:
Let $$H(p) := - p \log p - (1 - p) \log (1- p)$$
be the entropy of a Bernoulli variable with success probability $p$. One could ask when the entropy of a single coin flip with head probability $p$ is equal to the entropy of 2 iid coin flips of the same head probability. In other words, solve $p$ in
$$H(p) = H(1 - (1-p)^2) = H(2p - p^2)$$.
It turns out that $p^* = \frac{3 - \sqrt{5}}{2}$ is the unique solution between $0$ and $1$. The quantity $H(p) - H(2p - p^2)$ however is a linear combination of $\log p$, $\log (1-p)$ and $\log(2 - p)$, with coefficients polynomial in $p$, so a priori there should not be algebraic solutions: indeed, Wolfram is not able to solve it. To verify it, one needs $p^* = (1 - p^*)^2$ and $(1 - p^*) ( 2 - p^*) = 1$.
So why is there such a coincidence? Note that this fact is not a crucial step in Gilmer's proof: approximate numeric solution suffices.