AlphaEvolve problems·alphaevolve:29

Block Stacking Problem

Let $n \\geq 1$. Let $C(n)$ be the largest displacement that the $n^{\\mathrm{th}}$ block in a stack of identical rigid rectangular blocks of width $1$ can be displaced horizontally over the edge of a table, with the stack remaining stable. More mathematically, $C(n)$ is the supremum of $x_n$ where $0 = x_0 \\leq x_1 \\leq \\dots\\leq x_n$ are real numbers subject to the constraints $$ \\frac{x_{i+1}+\\dots+x_n}{n-i} < x_i + \\frac{1}{2}$$ for all $0 \\leq i < n$. What is $C(n)$?

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