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The payoff in the Chow-Robbins coin-tossing game is the proportion of headswhen you stop. Knowing when to stop to maximize expectation was addressed byChow and Robbins(1965), who proved there exist integers ${k_n}$ such that it isoptimal to stop when heads minus tails reaches this. Finding ${k_n}$ exactlywas unsolved except for finitely many cases by computer. We show ${k_n} =\left\lceil {\alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2\zeta ( -1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}}} \right\rceil$ foralmost all n, where $\alpha $ is the Shepp-Walker constant.This comes from ourestimate ${\beta_n} = \alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { -2\zeta ( - 1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}} + O\left({{n^{ - 7/24}}} \right)$ of real numbers defined by Dvoretzky(1967) for a moregeneral Value function which is continuous in its first argument and easier toanalyze. An $O({n^{ - 1/4}})$ dependence was conjectured by Christensen andFischer(2022) from numerical evidence. Our proof uses moments involving Catalanand Catalan triangle numbers which appear in a tree resulting from backwardinduction, and a generalized backward induction principle. It was motivated byan idea of H\"aggstr\"om and W\"astlund(2013) to use backward induction ofupper and lower Value bounds from a horizon, which they used numerically tosettle a few cases. Christensen and Fischer, with much better bounds, settledmany more cases. We use Skorohod's embedding to get simple upper and lowerbounds from the Brownian analog; our upper bound is the one found byChristensen and Fischer in a different way. We use them first for many moreexamples, but the new idea is to use them algebraically in the tree, withfeedback to get a sharper Value estimate near the border, to settle almost alln.

Exact Solution to the Chow-Robbins Game for almost all n, using the Catalan Triangle