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So Long Sucker: Endgame Analysis

Jean-Lou De Carufel, Marie Rose Jerade

TL;DR

This paper tackles the endgame analysis of So Long Sucker, focusing on the two-player two-color scenario (Blue vs Red). It develops a systematic, inductive, case-analysis framework over board types (Type I, generalized Type I, Type II, generalized Type II) and introduces quantitative thresholds (e.g., $m_b>n_r$ and its generalized variants) that determine when the active player has a winning strategy. The main result is a final theorem unifying these cases: Blue (the active player) has a winning strategy precisely when $m_b>0$ and either $n_r=0$ or $m_b+ extstyleig|eta_iig|_b > n_r+ extstyleig| ho_iig|_r- ext{max}_iig| ho_iig|_r$, with the strategy $ ext{S}$ providing the construction. This advances the mathematical understanding of SLS endgames and offers a framework for exploring more colors and potential equilibrium concepts in cooperative-competitive dynamics.

Abstract

So Long Sucker is a strategy board game that requires 4 players, each with $c$ chips of their designated color, and a board made of $k$ empty piles. With a clear set-up comes intricate rules, such as: players taking turns but not in a fixed order, agreements made between some players broken at any time, or a player winning the game without any chips in hand. One of the main points of interest in studying this game is finding when a player has a winning strategy. The game begins with four players who get successively eliminated until only the winner is left. To study winning strategies, it is of interest to look at endgame situations. For that, we study the following game set-up: there are two players left in the game, Blue and Red, with only their respective chip colors. In this paper, we characterize Blue's winning scenarios and strategies for this game set-up through a delicate case analysis.

So Long Sucker: Endgame Analysis

TL;DR

This paper tackles the endgame analysis of So Long Sucker, focusing on the two-player two-color scenario (Blue vs Red). It develops a systematic, inductive, case-analysis framework over board types (Type I, generalized Type I, Type II, generalized Type II) and introduces quantitative thresholds (e.g., and its generalized variants) that determine when the active player has a winning strategy. The main result is a final theorem unifying these cases: Blue (the active player) has a winning strategy precisely when and either or , with the strategy providing the construction. This advances the mathematical understanding of SLS endgames and offers a framework for exploring more colors and potential equilibrium concepts in cooperative-competitive dynamics.

Abstract

So Long Sucker is a strategy board game that requires 4 players, each with chips of their designated color, and a board made of empty piles. With a clear set-up comes intricate rules, such as: players taking turns but not in a fixed order, agreements made between some players broken at any time, or a player winning the game without any chips in hand. One of the main points of interest in studying this game is finding when a player has a winning strategy. The game begins with four players who get successively eliminated until only the winner is left. To study winning strategies, it is of interest to look at endgame situations. For that, we study the following game set-up: there are two players left in the game, Blue and Red, with only their respective chip colors. In this paper, we characterize Blue's winning scenarios and strategies for this game set-up through a delicate case analysis.
Paper Structure (23 sections, 27 theorems, 53 equations, 7 figures, 1 table)

This paper contains 23 sections, 27 theorems, 53 equations, 7 figures, 1 table.

Table of Contents

  1. So Long Sucker: An Introduction
  2. Setup
  3. The Rules of the Game
  4. First Player:
  5. Active Player's Turn:
  6. Capture Rule:
  7. Next Player Rule:
  8. Donation Rule:
  9. Discarding Rule:
  10. Deal Rule:
  11. The Winner:
  12. Remarks
  13. Example
  14. Endgame Situations with Two Players and Two Colors
  15. Boards Without Long $b$-Piles
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $\mathscr{X} \in\{\mathscr{B}, \mathscr{R}\}$ be a player with designated color $x$ and let $y\in\{b,r\}$ be a color such that $y\neq x$. If $\mathscr{X}$ is the active player and makes any of the following three moves (assuming they have the appropriate chips in their possession), then they get

Figures (7)

  • Figure 1: SLS setup with $c=6$ and $k=8$. The symbol $\_$ illustrates the bottom of a pile.
  • Figure 2: Yellow is the active player.
  • Figure 3: Blue is the active player.
  • Figure 4: Red is the active player.
  • Figure 5: Red is the active player.
  • ...and 2 more figures

Theorems & Definitions (54)

  • Theorem 2.1: Same Active Player
  • proof
  • Theorem 2.2: Different Active Player
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5: Active Player's Round
  • Definition 2.6: Strategy $\mathcal{S}$
  • ...and 44 more