Puzzle—The Fillomino Puzzle

Abstract

Abstract. Logic puzzles form an excellent set of problems for the teaching of advanced solution techniques in operations research. They are an opportunity for students to test their modelling skills on a different style of problem, and some puzzles even require advanced techniques to become tractable. Fillomino is a puzzle in which the player must enter integers into a grid to satisfy certain rules. This puzzle is a good exercise in using lazy constraints and composite variables to solve difficult problems.

Puzzle—Solving Smartphone Puzzle Apps by Mathematical Programming

“…DePuy and Taylor (2007) address board puzzles such as the Cracker Barrel peg game and the rush hour problem. Pearce and Forbes (2017) discuss the Fillomino Puzzle. Yeomans (2003) examine what is usually called "Einstein's riddle;" Chlond (2014) consider other logic grid puzzles.…”

Knights Exchange Puzzle—Teaching the Efficiency of Modeling

“…Then, they improved the solution time significantly by reducing the number of variables and constraints, as well as introducing short subtour elimination constraints. Furthermore, Pearce and Forbes (2017) considered the Fillomino puzzle, which is difficult to formulate as a complete mixed integer programming model. They developed two incomplete formulations using lazy constraints and composite variables.…”

Towards a general formulation of lazy constraints

“…The composite variables formulation for the Fillomino puzzle is an example of this [1]. To build as many constraints into the variables as possible, each variable would represent the placement of a tile and the values of all cells neighbouring the tile.…”

“…We are only considering the scheduling constraints (MIP3-MIP4) here. When the variables y at and z rt are allowed to be continuous, it is possible for z rt to take values of1 3 at times 0, 10 and 20, and 0 elsewhere. Because in constraint (MIP4) we sum the z rt over values of t within Durr time periods previous to t, at every t ∈ [Re r ,De r ], ∑ r∈R a min{t,De r } ∑ t =max{Re r ,t−Dur r +1} z rt = 1 3 .…”

“…1 shows a comparison of three formulations on 10 randomly generated instances with 50 cities each. The locations are uniformly distributed in a square area and the distances between locations are euclidean.…”