Mr. Markov Aviates Aeroplane Chess

Aeroplane Chess “飞行棋” is an East Asian board game in which each player commands a fleet of four planes that race around the board to finish at the final landing spot, in the center of the game board. Players take turns advancing their planes by rolling an ordinary six-sided die. Play ends when one player has moved all four of their planes to the final landing spot [Asiapac Editorial 2011, 65–67]. The game thus resembles Ludo, Parcheesi, and backgammon.

Aeroplane Chess is a chance-based game that under simplified conditions allows a Markov chain analysis. The analysis here is similar to those of three other common childhood board games: Chutes and Ladders [Gadbois 1993], Candy Land [Mack 2014], and Goose [Puri 2020].

We model a single player’s plane moving around the game board. Restricting to one plane, rather than the full four, removes any choice on the player’s part of which plane to advance, so that a Markov chain methodology can be used. The creation of a transition matrix will yield the expected number of moves to finish from any square (most importantly, from the hangar at the start) and the hidden structures of the board, paying attention to jump rules, shortcuts, and other key features of the board.

Note: The information below was created with the assistance of AI.

Level of Mathematics

Overall Level:
Upper High School - Early Undergraduate (Introductory College Level)

Evidence:

• Uses Markov chains, transition matrices, and expected values
• Includes:
• Matrix algebra (55×55 matrices)
• Probability distributions and expected value formulas
• Inverse matrices

Interpretation:

• Accessible to:
• Strong high school students (AP Statistics / advanced math)
• Intro college students in:
• Probability
• Linear algebra
• Discrete math

Subject Matter
Primary Mathematical Topics:

• Probability Theory
• Discrete probability
• Probability mass functions
• Markov Chains / Stochastic Processes
• Transition matrices
• Absorbing states
• Linear Algebra
• Matrix construction and inversion
• Expected Value & Distributions
• Expected number of moves
• Distribution of game length

Secondary Topics:

• Simulation concepts (implicit)
• Game theory (light, informal)
• Combinatorics (dice outcomes)

Application Areas
Core Application:

• Game Analysis / Board Games
• Aeroplane Chess modeled mathematically

Broader Real-World Applications:

• Stochastic modeling
• Systems with random transitions
• Operations research
• Expected time to completion
• Computer science
• Random processes / state machines
• Data science
• Transition probabilities and distributions

Educational Application:

• Demonstrates how real-world systems → mathematical models

Prerequisites
Required Background Knowledge:

Core:

• Basic probability:
• Events, outcomes, independence
• Algebra:
• Equations, summations
• Matrices (intro level)

Recommended:

• Understanding of:
• Expected value
• Conditional probability
• Discrete distributions

Advanced (helpful but not strictly required):

• Matrix inversion
• Markov chains (conceptual familiarity)

Correlation to Mathematics Standards
US Alignment with Common Core (High School):

Statistics & Probability

• HSS-CP: Conditional probability
• HSS-MD: Modeling with probability
• HSS-IC: Making inferences

Strong alignment:

• Probability distributions
• Expected values
• Modeling random processes

Alignment with AP Courses:

AP Statistics

• Probability distributions
• Expected value
• Simulation and modeling

AP Calculus (Indirect)

• Series notation appears (summations), but calculus is not central

Undergraduate Standards (Typical Courses):

• Intro Probability / Statistics
• Discrete Mathematics
• Linear Algebra (applications)
• Stochastic Processes (intro level)

Cognitive / Mathematical Practices (Standards for Mathematical Practice)

This paper strongly reflects:

• MP2: Reason abstractly and quantitatively
• MP4: Model with mathematics
• MP7: Look for structure
• MP8: Repeated reasoning (matrix powers, distributions)