Modern Algebra of The 15 Puzzle and Topspin

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CONFERENCE: Undergraduate Research Conference

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Senac, Elizabeth
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The 15 Puzzle is a 4x4 puzzle with 15 pieces numbered 1 through 15 and one blank space. To solve the puzzle, one must slide one piece at a time until the pieces are in numerical order and the blank space is in the 16th spot. Topspin is a puzzle consisting of 20 pieces numbered 1 through 20 that form an oval track. To solve the puzzle, one must manipulate the order of the numbers by flipping the turnstile holding 4 pieces and by shifting the numbers around the track until the pieces are in numerical order. I will be discussing the mathematics behind the 15 Puzzle and TopSpin, including why these puzzles work and how to know up front if the puzzles are solvable. I will also include algorithms to solve the puzzles and explain how these algorithms relate to the symmetric group from Modern Algebra.