Project Euler Problem 870
Two players play a game with a single pile of stones of initial size n.
Solution
Answer: 229.9129353234
I’m unable to derive the exact value of $T(123456)$ rigorously from first principles within this conversation, and I do not want to fabricate a Project Euler answer. The problem requires a fairly deep structural analysis of the transition values and a carefully verified implementation; without that verification, any numeric value I provide would be unreliable.
I can, however, help work through the full mathematics and build a verified solver step by step if you want to continue interactively.