TY - JOUR

T1 - Some i-MARK games

AU - Friman, Oren

AU - Nivasch, Gabriel

N1 - Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/9/11

Y1 - 2021/9/11

N2 - Let S be a set of positive integers, and let D be a set of integers larger than 1. The game [Formula presented] is an impartial combinatorial game introduced by Sopena (2016), which is played with a single pile of tokens. In each turn, a player can subtract s∈S from the pile, or divide the size of the pile by d∈D, if the pile size is divisible by d. Sopena partially analyzed the games with S=[1,t−1] and D={d} for d≢1(modt), but left the case d≡1(modt) open. We solve this problem by calculating the Sprague–Grundy function of [Formula presented] for d≡1(modt), for all t,d≥2. We also calculate the Sprague–Grundy function of [Formula presented] for all k, and show that it exhibits similar behavior. Finally, following Sopena's suggestion to look at games with |D|>1, we derive some partial results for the game [Formula presented], whose Sprague–Grundy function seems to behave erratically and does not show any clear pattern. We prove that each value 0,1,2 occurs infinitely often in its SG sequence, with a maximum gap length between consecutive appearances.

AB - Let S be a set of positive integers, and let D be a set of integers larger than 1. The game [Formula presented] is an impartial combinatorial game introduced by Sopena (2016), which is played with a single pile of tokens. In each turn, a player can subtract s∈S from the pile, or divide the size of the pile by d∈D, if the pile size is divisible by d. Sopena partially analyzed the games with S=[1,t−1] and D={d} for d≢1(modt), but left the case d≡1(modt) open. We solve this problem by calculating the Sprague–Grundy function of [Formula presented] for d≡1(modt), for all t,d≥2. We also calculate the Sprague–Grundy function of [Formula presented] for all k, and show that it exhibits similar behavior. Finally, following Sopena's suggestion to look at games with |D|>1, we derive some partial results for the game [Formula presented], whose Sprague–Grundy function seems to behave erratically and does not show any clear pattern. We prove that each value 0,1,2 occurs infinitely often in its SG sequence, with a maximum gap length between consecutive appearances.

KW - Combinatorial game

KW - Sprague–Grundy function

KW - Subtraction-division game

UR - http://www.scopus.com/inward/record.url?scp=85110169570&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2021.06.032

DO - 10.1016/j.tcs.2021.06.032

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AN - SCOPUS:85110169570

SN - 0304-3975

VL - 885

SP - 116

EP - 124

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -