arXiv subjects

Urban Larsson

Publications and source records attributed to Urban Larsson.

At least 19 recordsLinked to original sources

arXiv2026

Partizan Subtraction with Full and Truncated Support

We investigate {\sc Full Support (FS)}, a {\sc Partizan Subtraction} game in which the players can remove any number of pebbles from the heap up to certain bounds that are typically different for Left and Right. The player with the richer move set always wins for all but finitely many heap sizes. We confirm this advantage by finding the general canonical form and the atomic weights of this game. To restore fairness (and peace), we introduce {\sc Truncated Support (TS)}, which essentially trims the larger subtraction set from below. If the truncation is shallow, the unfairness persists above a certain heap size, and one player continues ruling. If the truncation is deep, another player starts ruling. Interestingly, there is one more balanced truncation level in the middle, for which a non-trivial periodicity emerges, and where it has infinitely many $\mathcal P$ and $\mathcal N$-positions. We also explore the atomic weights for the lightly trimmed scenarios.

math.CO↗

arXiv2026

A Subtraction Nim with a Pass

We consider a subtraction Nim with subtraction set {s_1,s_2,s_3={2,4n,4n+2}, where n is a positive integer such that n >= 3. We do not treat the case that n=1 or n=2 in this article. We show that this game satisfies the reverse-mex property of Grundy numbers, i.e., G(x)=mex{G(x+s_1), G(x+s_2), G(x+s_3)}, where the mex is taken over successors rather than predecessors. We modify the rule of this subtraction Nim to allow a one-time pass, that is, a passing move usable at most once during the game, unavailable from terminal positions; once used by either player, it becomes unavailable. In classical Nim, the introduction of a pass move complicates the game, and finding a formula that describes the set of P-positions in traditional three-pile Nim with a pass remains an important open question. In the case of subtraction Nim with a pass, however, the introduction of a pass move does not complicate the game. We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available.

math.CO↗

arXiv2026

Additive Subtraction Games

We determine the full nim-value structure of additive subtraction games in the {\em primitive quadratic} regime. The problem appears in Winning Ways by Berlekamp et al. in 1982; it includes a closed formula, involving Beatty-type {\em bracket expressions} on rational moduli, for determining the {\mathscr P}-positions, but to the best of our knowledge, a complete proof of this claim has not yet appeared in the literature; Mikl\'os and Post (2024) established outcome-periodicity, but without reference to that closed formula. The {\em primitive quadratic} case captures the source of the quadratic complexity of the problem, a claim supported by recent research in the dual setting of sink subtraction by Bhagat et al. This study focuses on a number theoretic solution involving the classical closed formula, and we establish that each nim-value sequence resides on a linear shift of the classical {\mathscr P}-positions.

math.CO↗

arXiv2026

Additive sink subtraction

Subtraction games are a classical topic in Combinatorial Game Theory. A result of Golomb~(1966) shows that every subtraction game with a finite move set has an eventually periodic nim-sequence, but the known proof yields only an exponential upper bound on the period length. Flammenkamp~(1997) conjectures a striking classification for three-move subtraction games: non-additive rulesets exhibit linear period lengths of the form ``the sum of two moves'', where the choice of which two moves displays fractal-like behavior, while additive sets $S=\{a,b,a+b\}$ have purely periodic outcomes with linear or quadratic period lengths. Despite early attention in Winning Ways~(1982), the general additive case remains open. We introduce and analyze a dual winning convention, which we call {\sc sink subtraction}. Unlike the standard {\em wall} convention, where moves to negative positions are forbidden, the sink convention declares a player the winner upon moving to a non-positive position. We show that {\sc additive sink subtraction} admits a complete solution: the nim-sequence is purely periodic with an explicit linear or quadratic period formula, and we conjecture a duality between additive sink subtraction and classical wall subtraction. Keywords: Additive Subtraction Game, Nimber, Periodicity, Sink Convention.

math.CO↗

arXiv2025

Cyclic impartial games with carry-on moves

In an impartial combinatorial game, both players have the same options in the game and all its subpositions. The classical Sprague-Grundy Theory was developed for short impartial games, where players have a finite number of options, there are no special moves, and an infinite run is not possible. Subsequently, many generalizations have been proposed, particularly the Smith-Frankel-Perl Theory devised for games where the infinite run is possible, and the Larsson-Nowakowski-Santos Theory able to deal with entailing moves that disrupt the logic of the disjunctive sum. This work presents a generalization that combines these two theories, suitable for analyzing cyclic impartial games with carry-on moves, which are particular cases of entailing moves where the entailed player has no freedom of choice in their response. This generalization is illustrated with sc green-lime hackenbush, a game inspired by the classic green hackenbush.

math.CO↗

arXiv2025

Tie-breaking in self interest cumulative subtraction games

Subtraction games have a rich literature as normal-play combinatorial games (e.g., Berlekamp, Conway, and Guy, 1982). Recently, the theory has been extended to zero-sum scoring play (Cohensius et al. 2019). Here, we take the approach of cumulative self-interest games, as introduced in a recent framework preprint by Larsson, Meir, and Zick. By adapting standard Pure Subgame Perfect Equilibria (PSPE) from classical game theory, players must declare and commit to acting either ``friendly'' or ``antagonistic'' in case of indifference. Whenever the subtraction set has size two, we establish a tie-breaking rule monotonicity: a friendly player can never benefit by a deterministic deviation to antagonistic play. This type of terminology is new to both ``economic'' and ``combinatorial'' games, but it becomes essential in the self-interest cumulative setting. The main result is an immediate consequence of the tie-breaking rule's monotonicity; in the case of two-action subtraction sets, two antagonistic players are never better off than two friendly players, i.e., their PSPE utilities are never greater. For larger subtraction sets, we conjecture that the main result continues to hold, while tie-breaking monotonicity may fail, and we provide empirical evidence in support of both statements.

math.CO↗

arXiv2025

Thresholds of Queen covers

We study optimal configurations of Queens on a square chessboard, defined as those covering the maximum number of squares. For a fixed number of Queens, $q$, we prove the existence of two thresholds in board size: a non-attacking threshold beyond which all optimal configurations are pairwise non-attacking, and a stabilizing threshold beyond which the set of optimal configurations becomes constant. Related studies on Queen domination, such as Tarnai and G\'asp\'ar (2007), focus on minimizing the number of Queens needed for full board coverage. Our approach, by contrast, fixes the number of Queens and analyzes optimal cover via a certain loss-function due to {\em internal loss} and {\em decentralization}. We demonstrate how the internal loss can be decomposed in terms of defined concepts, {\em balance} and {\em overlap concentration}. Moreover, by using our results, for sufficiently large board sizes, we find all optimal Queen configurations for all $2\le q\le 9$. And, whenever possible, we relate those solutions in terms of the classical problem of placing $q$ non-attacking Queens on a $q\times q$ board. For example, in case $q=8$, out of the twelve classical fundamental solutions, only three apply here as centralized patterns on large boards. On the other hand, the single classical fundamental solution for $q=6$ is never cover optimal on large boards, even if centralized, but another pattern that fits inside a $q\times (q+1)$ board applies.

math.CO↗

arXiv2025

A number game reconciliation

Number games play a central role in alternating normal play combinatorial game theory due to their real-number-like properties (Conway 1976). Here we undertake a critical re-examination: we begin with integer and dyadic games and identify subtle inconsistencies and oversights in the established literature (e.g. Siegel 2013), most notably, the lack of distinction between a game being a number and a game being equal to a number. After addressing this, we move to the general theory of number games. We analyze Conway's original definition and a later refinement by Siegel, and highlight conceptual gaps that have largely gone unnoticed. Through a careful dissection of these issues, we propose a more coherent and robust formulation. Specifically, we develop a refined characterization of numbers, via several subclasses, dyadics, canonical forms, their group theoretic closure and zugzwangs, that altogether better capture the essence of number games. This reconciliation not only clarifies existing ambiguities but also uncovers several open problems.

cs.GT↗

arXiv2025

Temperatures of Robin Hood

Cumulative Games were introduced by Larsson, Meir, and Zick (2020) to bridge some conceptual and technical gaps between Combinatorial Game Theory (CGT) and Economic Game Theory. The partizan ruleset {\sc Robin Hood} is an instance of a Cumulative Game, viz., {\sc Wealth Nim}. It is played on multiple heaps, each associated with a pair of cumulations, interpreted here as wealth. Each player chooses one of the heaps, removes tokens from that heap not exceeding their own wealth, while simultaneously diminishing the other player's wealth by the same amount. In CGT, the {\em temperature} of a {\em disjunctive sum} game component is an estimate of the urgency of moving first in that component. It turns out that most of the positions of {\sc Robin Hood} are {\em hot}. The temperature of {\sc Robin Hood} on a single large heap shows a dichotomy in behavior depending on the ratio of the wealths of the players. Interestingly, this bifurcation is related to Pingala (Fibonacci) sequences and the Golden Ratio $ϕ$: when the ratio of the wealths lies in the interval $(ϕ^{-1},ϕ)$, the temperature increases linearly with the heap size, and otherwise it remains constant, and the mean values has a reciprocal property. It turns out that despite {\sc Robin Hood} displaying high temperatures, playing in the hottest component might be a sub-optimal strategy.

math.CO↗

arXiv2024

Enforce and selective operators of combinatorial games

We consider an {\em enforce operator} on impartial rulesets similar to the Muller Twist and the comply/constrain operator of Smith and St\u anic\u a, 2002. Applied to the rulesets A and B, on each turn the opponent enforces one of the rulesets and the current player complies, by playing a move in that ruleset. If the outcome table of the enforce variation of A and B is the same as the outcome table of A, then we say that A dominates B. We find necessary and sufficient conditions for this relation. Additionally, we define a {\em selective operator} and explore a distributive-lattice-like structure within applicable rulesets. Lastly, we define nim-values under enforce-rulesets, and establish that the Sprague-Grundy theory continues to hold, along with illustrative examples.

math.CO↗

arXiv2024

A brief conversation about subtraction games

In this survey we revisit {\sc finite subtraction}, one-heap subtraction games on finite rulesets. The main purpose is to give a general overview of the development, and specifically to draw attention to Flammenkamp's thesis (1997), where he, contrary to other studies, experimentally observes exponential eventual period length of the outcomes, for a carefully selected subclass of games. In addition, we contribute an appendix on {\sc finite excluded subtraction} by Suetsugu.

math.CO↗

arXiv2024

Affine Normal Play

There are many combinatorial games in which a move can terminate the game, such as a checkmate in chess. These moves give rise to diverse situations that fall outside the scope of the classical normal play structure. To analyze these games, an algebraic extension is necessary, including infinities as elements. In this work, affine normal play, the algebraic structure resulting from that extension, is analyzed. We prove that it is possible to compare two affine games using only their forms. Furthermore, affine games can still be reduced, although the reduced forms are not unique. We establish that the classical normal play is order-embedded in the extended structure, constituting its substructure of invertible elements. Additionally, as in classical theory, affine games born by day n form a lattice with respect to the partial order of games.

math.CO↗

arXiv2024

Subtraction games in more than one dimension

This paper concerns two-player alternating play combinatorial games (Conway 1976) in the normal-play convention, i.e. last move wins. Specifically, we study impartial vector subtraction games on tuples of nonnegative integers (Golomb 1966), with finite subtraction sets. In case of two move rulesets we find a complete solution, via a certain $\mathcal{P}$-to-$\mathcal{P}$ principle (where $\mathcal{P}$ means that the previous player wins). Namely $x \in \mathcal{P}$ if and only if $x +a +b \in \mathcal{P}$, where $a$ and $b$ are the two move options. Flammenkamp 1997 observed that, already in one dimension, rulesets with three moves can be hard to analyze, and still today his related conjecture remains open. Here, we solve instances of rulesets with three moves in two dimensions, and conjecture that they all have regular outcomes. Through several computer visualizations of outcomes of multi-move two-dimensional rulesets, we observe that they tend to partition the game board into periodic mosaics on very few regions/segments, which can depend on the number of moves in a ruleset. For example, we have found a five-move ruleset with an outcome segmentation into six semi-infinite slices. In this spirit, we develop a coloring automaton that generalizes the $\mathcal{P}$-to-$\mathcal{P}$ principle. Given an initial set of colored positions, it quickly paints the $\mathcal{P}$-positions in segments of the game board. Moreover, we prove that two-dimensional rulesets have row/column eventually periodic outcomes. We pose open problems on the generic hardness of two-dimensional rulesets; several regularity conjectures are provided, but we also conjecture that not all rulesets have regular outcomes.

math.CO↗

arXiv2023

Memgames

In this article, we study the structure, and in particular the Grundy values, of a family of games known as memgames.

math.CO↗

arXiv2023

Bidding combinatorial games

Combinatorial Game Theory is a branch of mathematics and theoretical computer science that studies sequential 2-player games with perfect information. Normal play is the convention where a player who cannot move loses. Here, we generalize the classical alternating normal play to infinitely many game families, by means of discrete Richman auctions (Develin et al. 2010, Larsson et al. 2021, Lazarus et al. 1996). We generalize the notion of a perfect play outcome, and find an exact characterization of outcome feasibility. As a main result, we prove existence of a game form for each such outcome class; then we describe their lattice structures. By imposing restrictions to the general families, such as impartial and {\em symmetric termination}, we find surprising analogies with alternating play.

cs.GT↗

arXiv2023

Constructive comparison in bidding combinatorial games

A class of discrete Bidding Combinatorial Games that generalize alternating normal play was introduced by Kant, Larsson, Rai, and Upasany (2022). The major questions concerning optimal outcomes were resolved. By generalizing standard game comparison techniques from alternating normal play, we propose an algorithmic play-solution to the problem of game comparison for bidding games. We demonstrate some consequences of this result that generalize classical results in alternating play (from Winning Ways 1982 and On Numbers and Games 1976). In particular, integers, dyadics and numbers have many nice properties, such as group structures, but on the other hand the game * is non-invertible. We state a couple of thrilling conjectures and open problems for readers to dive into this promising path of bidding combinatorial games.

cs.GT↗

arXiv2023

A complete solution for a nontrivial ruleset with entailing moves

Combinatorial Game Theory typically studies sequential rulesets with perfect information where two players alternate moves. There are rulesets with {\em entailing moves} that break the alternating play axiom and/or restrict the other player's options within the disjunctive sum components. Although some examples have been analyzed in the classical work Winning Ways, such rulesets usually fall outside the scope of the established normal play mathematical theory. At the first Combinatorial Games Workshop at MSRI, John H. Conway proposed that an effort should be made to devise some nontrivial ruleset with entailing moves that had a complete analysis. Recently, Larsson, Nowakowski, and Santos proposed a more general theory, {\em affine impartial}, which facilitates the mathematical analysis of impartial rulesets with entailing moves. Here, by using this theory, we present a complete solution for a nontrivial ruleset with entailing moves.

math.CO↗

arXiv2021

Absolute Combinatorial Game Theory

We propose a unifying additive theory for standard conventions in Combinatorial Game Theory, including normal-, misère- and scoring-play, studied by Berlekamp, Conway, Dorbec, Ettinger, Guy, Larsson, Milley, Neto, Nowakowski, Renault, Santos, Siegel, Sopena, Stewart (1976-2019), and others. A game {\em universe} is a set of games that satisfies some standard closure properties. Here, we reveal when the fundamental game comparison problem, ``Is $G\su H$?'', simplifies to a constructive `local' solution, which generalizes Conway's foundational result in ONAG (1976) for normal-play games. This happens in a broad and general fashion whenever a given game universe is {\em absolute}. Games in an absolute universe satisfy two properties, dubbed {\em parentality} and {\em saturation}, and we prove that the latter is implied by the former. Parentality means that any pair of non-empty finite sets of games is admissible as options, and saturation means that, given any game, the first player can be favored in a disjunctive sum. Game comparison is at the core of combinatorial game theory, and for example efficiency of potential reduction theorems rely on a local comparison. We distinguish between three levels of game comparison; superordinate (global), basic (semi-constructive) and subordinate (local) comparison. In proofs, a sometimes tedious challenge faces a researcher in CGT: in order to disprove an inequality, an explicit distinguishing game might be required. Here, we explain how this job becomes obsolete whenever a universe is absolute. Namely, it suffices to see if a pair of games satisfies a certain Proviso together with a Maintenance of an inequality.

math.CO↗