This riddle is a follow-up, with a novel twist, to the classic problems
from last month's puzzle. (Readers will do well to
tackle the previous challenge first. This month's riddle and its subsequent
solution are spoilers to last month's solution.)
Consider the following generalization to last month's "Part 4":
Once again, we refer to a turn-based game where two players, taking
alternate turns, each removes, in her turn, a chosen number of
elements from a pile. The initial pile-size is
On the first turn, the first to play can take any integer amount that is less
than The player to lose is the first to have no legal moves left. (For example, if the pile is completely emptied.)
Let
The question: characterize the limit of To receive credit, describe, as a function of α, a polynomial (other than the zero polynomial) that has this value as one of its roots (or prove that no such polynomial exists). Bonus mentions will be given to solvers who do this with a low-degree polynomial. As usual, all answers should be accompanied by proofs. |
## List of solvers:Radu-Alexandru Todor (*) (19 February 11:58)Liubing Yu (*) (19 February 18:51) Thomas Mack (28 February 03:16) |

Elegant and original solutions can be submitted to the puzzlemaster at __riddlesbrand.scso.com__.
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The solution will be published at the end of the month.

Enjoy!