Problem 870

Stone Game IV

Two players play a game with a single pile of stones of initial size $n$. They take stones from the pile in turn, according to the following rules which depend on a fixed real number $r>0$:

In the first turn, the first player may take $k$ stones with $1 \le k < n$.
If a player takes $m$ stones in a turn, then in the next turn the opponent may take $k$ stones with $1 \le k \le \lfloor r \cdot m \rfloor$.

Whoever cannot make a legal move loses the game.

Let $L(r)$ be the set of initial pile sizes $n$ for which the second player has a winning strategy. For example, $L(0.5) = \{1\}$, $L(1) = \{1, 2, 4, 8, 16, \dots\}$, $L(2) = \{1, 2, 3, 5, 8, \dots\}$.

A real number $q > 0$ is a transition value if $L(s)$ is different from $L(t)$ for all $s < q < t$.

Let $T(i)$ be the $i$-th transition value. For example, $T(1) = 1$, $T(2) = 2$, $T(22) \approx 6.3043478261$.

Find $T(123456)$ and give your answer rounded to $10$ digits after the decimal point.

取石子游戏（四）

两位玩家正在玩一个游戏，游戏开始时有一堆共$n$枚石子。两位玩家轮流从堆中取走石子，但必须遵循以下基于固定实数$r>0$的规则：

在第一轮中，先手玩家可以取走的石子数$k$需满足$1 \le k < n$。
如果一名玩家在某一轮取走$m$枚石子，则下一轮其对手可以取走的石子数$k$需满足$1 \le k \le \lfloor r \cdot m \rfloor$。

首先无法行动的玩家输掉游戏。

记$L(r)$为使得后手玩家有必胜策略的石子数$n$所构成的集合。例如，$L(0.5) = \{1\}$，$L(1) = \{1, 2, 4, 8, 16, \dots\}$，$L(2) = \{1, 2, 3, 5, 8, \dots\}$。

考虑实数$q>0$，若对于任意$s < q < t$均满足$L(s)$不等于$L(t)$，则称其为过渡值。
记$T(i)$为第$i$个过渡值。例如，$T(1) = 1$，$T(2) = 2$，$T(22) \approx 6.3043478261$。

求$T(123456)$，并四舍五入保留$10$位小数作为你的答案。

译注：“取石子游戏（三）”参见第366题。