Removing m from m-pile Divisor Nim
Abstract
In Divisor Nim, a player removes from one heap a positive number of stones that divides the size of every other heap. We prove that the Sprague--Grundy (SG) value satisfies g(P)≤ 2hmin+⌊log2hmin⌋, where hmin is the smallest nonempty heap, independently of the number and sizes of the other heaps. The proof starts with odd heaps and classifies removals by their factors of $2$. The largest heap in a starting board then supplies an SG bound valid throughout every successor chain. When one heap varies, we sharpen Morisawa's eventual-period bound: the smallest fixed heap controls the SG value range, the number of consecutive values compared, and an increase in the varying heap that repeats legal moves. The remaining dependence on the fixed board lies in the SG periods of direct successors. Using the largest fixed heap, we derive an explicit common period for all these recursive layers. Its possible prime divisors belong to a fixed finite set, independent of heap count; their exponents remain controlled by the total fixed stones. Finally, joining boards under one shared divisor rule preserves and can tighten the SG ceiling. Minimum 2-adic depth and its count parity determine every joined outcome, and a join with at least two odd heaps has exact SG value $0$ or $1$. We close with two conjectures: g(P)≤2hmin, and that the least eventual SG period, when one heap varies, divides 2lcm(1,…,hmax). Here hmax is the largest fixed heap, excluding the varying heap. The second claim would remove heap-count dependence from the period bound as well.
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BibTeX
@misc{tyagi2026removing,
title = {{Removing $m$ from $m$-pile Divisor Nim}},
author = {Satyam Tyagi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06925},
eprint = {2610.06925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06925},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Removing m from m-pile Divisor Nim
AU - Satyam Tyagi
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.06925
UR - https://arxiv.org/abs/2610.06925
AB - In Divisor Nim, a player removes from one heap a positive number of stones that divides the size of every other heap. We prove that the Sprague--Grundy (SG) value satisfies g(P)≤ 2h_min+⌊log_2h_min⌋, where h_min is the smallest nonempty heap, independently of the number and sizes of the other heaps. The proof starts with odd heaps and classifies removals by their factors of $2$. The largest heap in a starting board then supplies an SG bound valid throughout every successor chain. When one heap varies, we sharpen Morisawa's eventual-period bound: the smallest fixed heap controls the SG value range, the number of consecutive values compared, and an increase in the varying heap that repeats legal moves. The remaining dependence on the fixed board lies in the SG periods of direct successors. Using the largest fixed heap, we derive an explicit common period for all these recursive layers. Its possible prime divisors belong to a fixed finite set, independent of heap count; their exponents remain controlled by the total fixed stones. Finally, joining boards under one shared divisor rule preserves and can tighten the SG ceiling. Minimum 2-adic depth and its count parity determine every joined outcome, and a join with at least two odd heaps has exact SG value $0$ or $1$. We close with two conjectures: g(P)≤2h_min, and that the least eventual SG period, when one heap varies, divides 2lcm(1,…,h_max). Here h_max is the largest fixed heap, excluding the varying heap. The second claim would remove heap-count dependence from the period bound as well.
ER -