Puzzle - Chain-Reaction: A Puzzle to Demonstrate TSP Formulation

Introduction Many instructors use puzzles to help motivate and strengthen students’ integer programming (IP) formulation skills. The puzzle in this paper not only serves as another exercise in IP formulation but also provides an introduction to an often discussed OR topic: the traveling salesman problem (Dantzig et al. 1954). The chain-reaction puzzle is closely related to the traveling salesman problem (TSP) and as such allows students to discover an IP formulation for a TSP, including the need for subtour elimination constraints. Although more than one IP formulation of the TSP exists (Dantzig 1963, Miller et al. 1960, Svestka 1978), it has been this instructor’s experience that introductory OR students often formulate the chain-reaction puzzle using an assignment problem TSP formulation similar to that of Dantzig et al. (1954), Miller et al. (1960), and the one shown in the next section of this paper. The chain-reaction puzzle offers a nice introduction to the formulation and solution of the TSP because only a relatively small number of routes are feasible, thereby reducing the TSP to a manageable size for beginning students. Following this puzzle-based introduction to the TSP, the interested instructor is referred to Lee and Raffensperger (2006) for excellent TSP instructional materials. The chain-reaction puzzle is available on Andrea Gilbert’s website, http://clickmazes.com (Gilbert 1997). Figure 1 shows one possible instance of a 3 × 3 chain-reaction puzzle. The objective is to find a route through the puzzle board that visits each symbol exactly once. The next symbol in the route must match either the color or shape of the previous symbol. Moves need not be to adjacent cells; the route can leap around the board, but diagonal moves are not allowed. The starting cell is indicated by an “X,” but the ending cell is not given.