[Paper Review] The shortest game of Chinese Checkers and related problems

George I. Bell|ArXiv.org|Mar 8, 2008

Digital Image Processing Techniques2 references3 citations

TL;DR

This paper proves that the shortest possible two-player Chinese Checkers game is 30 moves (15 per player), confirming David Fabian's 1979 solution as optimal. Using A* and bidirectional search algorithms, the authors also establish that the minimal solitaire army transfer in Chinese Checkers requires 27 moves, matching Octave Levenspiel's 1971 solution and proving its optimality through computational verification under 6-move rules.

ABSTRACT

In 1979, David Fabian found a complete game of two-person Chinese Checkers in 30 moves (15 by each player) [Martin Gardner, Penrose Tiles to Trapdoor Ciphers, MAA, 1997]. This solution requires that the two players cooperate to generate a win as quickly as possible for one of them. We show, using computational search techniques, that no shorter game is possible. We also consider a solitaire version of Chinese Checkers where one player attempts to move her pieces across the board in as few moves as possible. In 1971, Octave Levenspiel found a solution in 27 moves [Ibid.]; we demonstrate that no shorter solution exists. To show optimality, we employ a variant of A* search, as well as bidirectional search.

Motivation & Objective

• To determine the minimum number of moves required to complete a two-player Chinese Checkers game under cooperative play.
• To verify the optimality of Octave Levenspiel's 1971 27-move solitaire solution for moving an army across a Chinese Checkers board.
• To apply computational search techniques to solve army transfer problems under different movement rules (4-move, 6-move, 8-move).
• To establish lower bounds and prove minimality of known solutions using algorithmic search and symmetry arguments.

Proposed method

• Employed A* search with heuristic functions based on centroid and distance-to-target to guide the search toward optimal solutions.
• Applied bidirectional search to reduce the search space by simultaneously expanding from the initial and goal states.
• Used a palindromic solution strategy, where solutions are mirrored and reversed to ensure completeness and symmetry.
• Defined movement rules precisely: 6-move (Chinese Checkers), 8-move (Halma), and 4-move (Checkers-like) rules on a 9×9 board.
• Implemented computational search on a 9×9 board with 10-man armies to verify minimal move counts.
• Used centroid-based lower bounds to estimate theoretical minimums, which were then matched by actual solutions.

Experimental results

Research questions

• RQ1 Is David Fabian’s 30-move solution the shortest possible game of two-player Chinese Checkers?
• RQ2 Can a solitaire army transfer in Chinese Checkers be completed in fewer than 27 moves?
• RQ3 What is the minimal number of moves required to transfer a 10-man army across a 9×9 board under 6-move rules?
• RQ4 How do different movement rules (4-move, 6-move, 8-move) affect the minimal solution length for army transfer problems?
• RQ5 Can computational search techniques like A* and bidirectional search be effectively used to prove optimality in combinatorial game puzzles?

Key findings

• The shortest possible two-player Chinese Checkers game is 30 moves, and no shorter game exists, confirming Fabian’s 1979 solution as optimal.
• The minimal solitaire solution for moving a 10-man army across the board under 6-move rules is 27 moves, and no shorter solution exists, proving Levenspiel’s 1971 solution optimal.
• For 8-move rules on a 9×9 board, the minimal army transfer requires 20 moves, and this solution is optimal.
• For 4-move rules, the minimal army transfer requires 30 moves, and this solution is optimal, with a specific sequence provided.
• A 19-man Halma army can be transferred across a 16×16 board in 47 moves, though this may not be the shortest possible.
• The paper introduces the concept of a 'balanced army' and observes that fastest solutions tend to be balanced, suggesting a direction for future research.