Martin J. Chlond

Puzzle---Three Set Pieces

In modeling integer and linear programs it is often the case that input data does not fit conveniently into regular array structures. In such cases it may be useful to employ notations taken from set theory that are implemented in many modeling languages. In this article three puzzle types are presented where the use of such notation leads to more general and elegant formulations than could be achieved without them.

Puzzle---IP Satisfied

Finding a feasible as opposed to optimal solution of an integer program model comes under the heading of constraint programming and if a model is also a binary integer program BIP it is known to be equivalent to the satisfiability SAT problem from Boolean logic. Many of the puzzles treated in this column over the years and a host of others fall into this category, that is, they may be modeled purely using binary variables and the aim is to fulfil the conditions of the puzzle rather than to optimize a function. This leads to the possibility that there may be gains in efficiency to be found by looking beyond general IP solvers to more specialized programs. The following article presents five puzzles together with IP formulations where the writer has experienced significant improvements in solution times using a SAT solver rather than a general purpose IP solver. The puzzles will be modeled and solved using three freely available open source packages, namely GLPK, Minisat and SCIP. GLPK and Minisat are both available in the GUSEK distribution from http://gusek.sourceforge.net/gusek.html and SCIP is available from http://scip.zib.de/ . GLPK is a general purpose linear and mixed integer solver, SCIP solves both Integer Programs and Constraint Programs and Minisat is exclusively an SAT solver. The GUSEK environment provides direct access to GLPK and Minisat. It also has the option to generate MPS formatted files of Mathprog models which may be read into SCIP and solved.

Puzzle-Logic Grid Puzzles

A type of logic-based puzzle, now referred to as a logic grid puzzle, began to appear in magazines sometime in the nineteen-eighties. In these puzzles the solver is provided with sets of attributes with an equal number of members in each set. The goal is to figure out which attributes are linked together based on a series of given clues. The attributes in each set are used once and only once and each puzzle has a unique solution which can be found using simple logic. A specific example of this type of puzzle is modeled and solved using integer programming and the lessons learned are used produce a set of generic rules that may be applied in solving a wide range of these puzzles.

Puzzle---Latin Square Puzzles

Latin squares and Graeco-Latin squares are mathematical objects much studied and loved by the recreational mathematics community. A Latin square is a n× n array of numbers where each row and column contains a single occurrence of each of the digits 1 n. A Graeco-Latin square, also known as an orthogonal Latin square, is a n× n array of cells where each cell contains an ordered pair (x y . The x and y values each form Latin squares and each cell contains a unique ordered pair. Integer program (IP) formulations of each of these structures are presented below. Puzzles consisting of Latin squares with additional constraints have been around for centuries and have become more popular in recent years. These include Sudoku and the “skyscrapers” puzzle, examples of which may be found at http://www .brainbashers.com. An integer programming approach to the former is described in Chlond (2005) and a description and IP formulation of the latter are included in this article. The 16 officers puzzle and the 36 cube puzzle are based on Graeco-Latin squares and are described and modeled in this article.

Puzzle—TSP at the Movies: Yondu’s Dart Problem

An unexpected application of the Traveling Salesman Problem (TSP) appears in the 2014 movie Guardians of the Galaxy. This is described and discussed and offered as a basis for a short student workshop to demonstrate the use of TSP.

Puzzle-Shedding Light on Higher Dimensions

Lights Out puzzles have been a popular computer pastime since the eighties. They consist of a grid of cells with each cell in one of two states, lit or unlit. When a cell is clicked, then that cell and its immediate neighbors toggle between lit and unlit. The objective is to identify the set of cells that must be clicked such that the entire grid is lit. Many variations on this theme devised by Jaap Scherphuis are at www.jaapsch.net/puzzles/lights.htm and one particularly interesting twist is at www.jaapsch.net/puzzles/java/lograph/lographapp.htm . This page implements, in the form of a Java applet, a set of tools to allow the creating, editing and playing of Lights Out puzzles on graphs. The nodes of the graph represent cells, and edges join any two cells that are considered neighbours. I show how to formulate Lights Out puzzles as integer programs and produce solutions using GLPK and Minisat. These are, respectively, open source software programs to solve Integer Programs and Satisfiability problems and are available within the GUSEK environment which may be downloaded from http://gusek.sourceforge.net/gusek.htm.

Puzzle-Chess Avoidance Puzzles

Atype of chess based puzzle devised by Erich Friedman is described and modeled using integer programming. In these puzzles the solver is required to place a given set of chess pieces on to an irregular grid such that no occupied squares are attacked by another piece. A MathProg model is included together with an Excel spreadsheet with the model embedded The purpose of this is to encourage instructors to investigate the possibility of using the Excel add-in SolverStudio as a teaching tool.

Puzzle-IP in the i

A couple of new and interesting puzzles have been appearing recently in the UK daily newspaper the i from the Independent. These are Wijuko and ABC Logic, both of which require Sudoku-like thinking to solve and may be modeled using Integer Programming IP. This article describes an example of each and develops IP models to generate solutions. The URL for the i website is http://www.independent.co.uk/i/ although this does not at present give access to the puzzles. The full version of the paper, including puzzles, may be viewed via http://www.pressreader.com/uk/i-from-the-independent/TextView .

Integer Programming in King Arthur's Court

In his quirky and whimsical book King Arthur in Search of His Dog by Smullyan (2010) the author presents the “Dogs of the Round Table” puzzle. Information regarding the number of King Arthur’s dogs that were at the Round Table is presented in a convoluted manner and the objective is to deduce the total number of dogs that earned this privilege. The reader is informed that each dog is either male or female, either shaggy or non-shaggy, and either large or small. There are 12 male dogs, 14 shaggy dogs, 13 large dogs, four large shaggy dogs, three large males, five shaggy males, one large shaggy male, and no small non-shaggy females. The puzzle boils down to a system of eight linear equations with eight variables. Two of these equations consist of variable assignments and the puzzle can be easily solved by back substitution. Nevertheless, the puzzle provides us with a neat student exercise in the visualization of a three-dimensional array and the use of sigma notation to identify subsets within such an array.