Huygens' solution to the gambler's ruin problem (Q1092882)

The fifth problem \((=\) the gambler's ruin problem) at the end of C. Huygens' treatise ``De ratiociniis in ludo aleae'' (1657) has been communicated only with its numerical result. The origin of the problem and its early solutions have been discussed by \textit{A. W. F. Edwards} [Int. Stat. Rev. 51, 73-79 (1983; Zbl 0502.60002)]. This note discusses Huygens' solution which has been found on a sheet of paper, dated 1676 [see Oeuvres complètes de C. Huygens, Vol. 14, 151-155 (1920). It contains an interesting diagrammatic component and indicates the structure of an essentially inductive algebraic argument using expectation trees.