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The Nature of Puzzles 1 Introduction 2 Puzzles As Games Article 1 The Nature of Puzzles Cameron Browne, QUT This paper explores the underlying nature of puzzles, and how they relate to games. The discussion focuses on pure deduction puzzles, but with reference to other types of puzzles where appropriate, with examples to support the concepts put forward. These include the notion of puzzles as two-player games between the setter and the solver, the addictiveness of puzzles, and ways in which the setter can exert authorial control, to make challenges more interesting and engaging for the solver. 1 Introduction zles, and observes that puzzles can be solved by pure reasoning alone, must have a complete analysis UZZLES come in many forms; there are word and that you are your own opponent, in the end [4]. P puzzles, jigsaw puzzles, logic puzzles, dex- While the first two observations are true in most terity puzzles, physical puzzles, physics based cases and they agree with most other definitions puzzles, to name just a few. While most readers of puzzles, I take issue with the third observa- will have an understanding of what the term ‘puz- tion that puzzles are a solitary pursuit undertaken zle’ means to them, the genre as a whole has so without an opponent. far defied exact definition, despite many attempts In his classic paper ‘Defining the Abstract’ (re- to do so. But perhaps a precise definition is not published in this issue [5]), Thompson makes the all that useful – or even possible – given the vari- astute observation that two-player abstract games ety of puzzles that exist. In this paper, I will look may be described as a series of puzzles that the instead at the underlying nature of puzzles rather players present to each other. Conversely, I be- than attempting to provide yet another definition. lieve that a puzzle may be described as a two- The central thesis of this paper is that most player game played between the setter and the puzzles are games played between the setter and solver. The task of the setter is to produce a chal- the solver, and that their inherent nature allows lenge that engages and entertains the solver, while sufficient authorial control for the setter to impart the task of the solver is to avoid the traps laid by their personality upon a well designed challenge, the setter to complete the challenge. in order to challenge, tease and engage the solver. It is worth emphasising that, unlike a player Several examples are presented in support of this in a traditional adversarial game, the setter is not argument, which are mostly taken from actual trying to ‘win’ against the solver. They are instead examples of pure deduction puzzles known as trying to provide the most entertaining playing Japanese logic puzzles [1]. These are characterised experience, in a role not dissimilar to that of the by having simple rules, a single (deducible) solu- gamemaster in a role playing game. In the lan- tion, and no language-dependent content. They guage of game theory, this is not a zero-sum game, are not only my favourite type of puzzle, but also as both players can win. illustrate the principles being discussed as clearly Puzzles are indeed solitary pursuits in a and simply as possible. strictly mathematical sense, as the solver is the To avoid confusion in the following discus- only agent making actions towards a solution. sion, the term puzzle will refer to the actual puzzle However, from a strategic or adversarial view- game itself, while each instance of a puzzle game point, these actions are directed by the informa- presented to the solver will be called a challenge. tion encoded in the challenge by the setter, which is revealed as the challenge unfolds. The solver 2 Puzzles As Games may not be an active player during the solution of a challenge, but participates in absentia by influ- A puzzle can be defined simply as a task that is encing the solver’s decisions and actions. fun and has a right answer [2], or more precisely: A well designed challenge will include traps a question which challenges people to solve, requires and deceptions posed by the setter, which the their deduction based on its rules to win, and doesn’t solver must detect and avoid. In order to see how depend on chance or other people’s action. [3] this works, let’s first look at the concepts of de- Schuh presents a classification scheme for puz- pendency and authorial control in puzzle design. Browne, C., ‘The Nature of Puzzles’, Game & Puzzle Design, vol. 1, no. 1, 2015, pp. 23–34. c 2015 2 Game & Puzzle Design Vol. 1, no. 1, 2015 21 8 21 5 8 219 5 8 5 1 2 9 5 1 2 9 5 1 2 9 9 3 2 4 9 3 2 4 9 3 2 4 8 5 6 8 5 6 8 5 6 6 7 6 7 6 7 3 1 8 3 1 8 3 1 8 8 7 1 6 8 7 1 6 8 7 1 6 6 9 5 1 6 9 5 1 6 9 5 1 4 2 4 2 4 2 (a) (b) (c) Figure 1. Dependent Sudoku hints progressively reveal enough information to make further progress. 2.1 Dependency the rules of Sudoku). The 1-hints provide enough information to instantiate another 1 on the top Dependency in this context refers to the degree row (a). This additional information allows a 5 to to which the steps required to solve a given chal- be instantiated in the same row (b), which in turn lenge are dependent on prior steps. A challenge in provides enough information to allow a 9 to be which progress can immediately be made at many instantiated next to it (c). places on the board shows low dependency, while The required information is meted out in in- a challenge that only exposes enough information stallments, in a self-perpetuating manner such for the solver to make progress at one particular that each action reveals further information to be point shows high dependency. The solver exploits acted upon. I have heard this process described as that piece of information... which reveals further the setter leaving a trail of (informational) bread information... which reveals further information... crumbs to follow. However, I prefer to think of until the solution is reached. the situation as a tapestry with a loose thread or Figure 1 shows a simple example of this pro- two, in which the majority of the position is im- cess in action, based on Sudoku challenge #31 penetrable except for certain weak points, whose from [6] (I assume that readers are familiar with exercise unravels further weak points to follow. 0 0 1 0 0 1 0 0 1 0 0 1 3 3 3 3 0 3 0 3 0 3 0 3 1 2 1 2 1 2 1 2 2 3 0 2 2 3 0 2 2 3 0 2 2 3 0 2 2 2 2 2 2 2 2 2 (a) (b) (c) (d) 0 0 1 0 0 1 0 0 1 0 0 1 3 3 3 3 0 3 0 3 0 3 0 3 1 2 1 2 1 2 1 2 2 3 0 2 2 3 0 2 2 3 0 2 2 3 0 2 2 2 2 2 2 2 2 2 (e) (f) (g) (h) Figure 2. An interesting section of a Slitherlink challenge. C. Browne The Nature of Puzzles 3 This Sudoku example only provides a superfi- perpetuating way. Each subproblem requires a cial instance of this process, as it is an easy chal- solution that provides the next subproblem, and lenge with several loose threads to follow. For so on. In a well designed puzzle, the solver can al- example, the 8-hints immediately dictate that the most feel the hand of the setter drip-feeding them central cell must take the number 8. Figure 2 information and leading them along by the nose shows a much more striking example, with Slith- along certain avenues to solution. erlink challenge #80 from [7]. Pelanek´ [8] describes the use of dependency as Slitherlink is a pure deduction puzzle in which a metric for automatically measuring the difficulty a simple closed path must be traced through or- of given Sudoku challenges, based on whether thogonal vertices of a square grid, to visit the num- the hints provide enough information to solve the ber of sides indicated on each numbered hint cell challenge in parallel (i.e. multiple loose threads [1]. Each adjacent pair of vertices therefore consti- to follow) or in series (i.e. narrow chain of depen- tutes a move whose value can either be an edge (j dent loose threads). I believe that dependency is or –) or no edge (×). a fundamental property of well designed puzzles Figure 2 (a) shows the lower left corner of the that runs deeper than just affecting difficulty, as initial challenge, and (b) shows three edges that it allows the setter to exert authorial control over must exist where a 3-hint meets a 0-hint. The path the challenges they construct. thus initiated then bounces off the 0-hints that it encounters (since the path can never visit the side 2.2 Authorial Control of a zero hint) and moves along the wall to give the position shown in Figure 2 (c). A deduction is Authorial control refers to the degree to which the required at this point; the dotted move can not be setter can influence the solver’s progress through an edge as that would cause the path to close pre- a given challenge, and manipulate their move maturely in a cross shape, so it must be no edge × choices in absentia.
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