What Is a Puzzle? A Definition From Mathematics and Puzzle Design

A puzzle is a problem with a fixed starting position, fixed rules and a clear goal, set so that you can reach the goal by reasoning alone. The sliding tile puzzle shows all three parts: the tiles start scrambled, you may only slide a tile into the empty space, and the goal is to put the tiles back in order ^[1].

Mathematicians who work on puzzles add two more ideas. A good puzzle has exactly one solution ^[3], and a hard puzzle is full of dead ends, even when only a single path leads to the goal ^[2]. This article looks at each of these ideas through peer-reviewed papers.

What do all puzzles have in common?

Look closely at almost any puzzle and you find the same three parts.

A starting position. A peg solitaire puzzle begins from a set pattern of pegs on a board of holes ^[2]. A sliding tile puzzle begins from a scrambled tray ^[1]. A dissection puzzle begins from a shape, such as a square, that you must cut up ^[4].

A set of rules. In peg solitaire, a move is one peg jumping over its neighbour into an empty hole, and the jumped peg is removed ^[2]. Jumps are allowed only along the lines of the board ^[2]. In a sliding tile puzzle, the only move is to slide a tile next to the empty space into it ^[1].

A goal. The goal of peg solitaire is to find a sequence of jumps that leaves exactly one peg on the board ^[2]. The goal of the sliding tile puzzle is to return every tile to its place ^[1]. The goal of a dissection puzzle is to cut one shape into pieces that can be rearranged into another shape of the same area ^[4].

The picture riddles on this site follow the same pattern. A picture riddle gives you a starting scene, a short set of rules, such as symbols keep their direction, and a single question to answer.

Are these ideas new?

Not at all. Peg solitaire has a history of about 300 years: it was invented in the late 17th century and set off an early craze for puzzles ^[2]. The knight's tour is older than the general mathematical question it belongs to: it predates an 1856 paper that posed that question for any network of points ^[5]. The mathematician Leonhard Euler presented knight's tours for the standard board of 8 rows and 8 columns ^[5].

What has changed is the tools. Today a computer can list every solvable position of a peg solitaire board and search that list for puzzles with exactly the properties a designer wants ^[2], and the same kind of search settled the 70 square ^[6].

What makes a puzzle a good one?

The clearest test comes from logic puzzles. A 2020 paper calls a puzzle good if it has exactly one solution, and clear if it can be solved from its given statements alone, with no outside information ^[3].

Both tests matter to a solver. A puzzle with two answers leaves you unsure whether you have finished. A puzzle that needs a hidden fact is not fair, because no amount of careful thinking will get you there.

The same paper shows how hard the first test can be to meet. In the simplest truth-teller and liar puzzles, where people only say who tells the truth and who lies, no puzzle that is both clear and good exists ^[3]. Puzzle makers have to add something extra, such as a count of truth-tellers or a new type of person, to reach a single answer ^[3].

What makes a puzzle hard?

A puzzle feels hard when there are many tempting moves and few that work. A 2017 paper on designing peg solitaire puzzles used a computer to look for exactly that ^[2]. It searched for starting positions where many first jumps are possible, but only one of them leads to a solution ^[2]. The author reasoned that to make a difficult puzzle, the number of dead ends should be large ^[2].

The search found puzzles that are much harder than the usual ones ^[2]. Many familiar puzzles, by contrast, start from a symmetric pattern, and those are often easy to solve by hand: a few matching jumps reduce the pattern to a smaller symmetric pattern that has already been solved ^[2].

A unique winning move also helps a clever solver. Once the single winning first jump is made, every jump that was already possible before it must still be a dead end, so the next jump has to be one the first jump opened up, and so on down the line ^[2]. A few of the puzzles found were even more special: they had only one full solution, which is rare in peg solitaire ^[2].

Size makes puzzles harder too. For the general sliding tile puzzle on a tray of any size, finding the shortest solution has been proved to be NP-hard, a mark of a very hard computing problem ^[1].

Can a puzzle have no solution at all?

Yes, and proving that is one of the most interesting things mathematicians do with puzzles.

On both of the 37-hole peg solitaire boards studied in a 2017 paper, filling every hole and removing only the centre peg gives a position that cannot be solved ^[2]. On the square-lattice board, the proof sorts every position into one of 16 classes, and a game can never leave the class it starts in; the hexagon board has four such classes ^[2].

Knight's tours give another example. A closed knight's tour visits every square of a board exactly once using knight moves and returns to its start ^[5]. Inside a cube built from stacked boards, a closed tour exists for cubes of side 4 or more exactly when the side is even ^[5]. Odd-sided cubes fail because of a colouring argument: the squares split into two colours that the knight must alternate between, and the two colours are not equal in number ^[5].

Dissections have their own limits. Thousands of dissections have been described, but proofs that a dissection uses the fewest possible pieces are rare ^[4]. A 2023 paper proved that a square and an equilateral triangle of the same area cannot be cut into three convex pieces that rearrange into each other ^[4]. So the famous four-piece solution, known as the Haberdasher Puzzle, is the best possible with convex pieces ^[4].

And the well-known 70 square puzzle, which asks whether squares of side 1 to 24 can exactly fill a square of side 70, has no solution either: the best packing leaves out the square of side 7 ^[6].

How do puzzles turn into mathematics?

Many puzzles become serious mathematics once someone asks a general question about them.

The knight's tour turns into a question about graphs, where each square is a point and each knight move is a line between two points ^[5]. Asking for a tour is the same as asking for a loop that passes through every point once, which mathematicians call a Hamiltonian cycle ^[5]. In 1991, a complete answer was given for which rectangular boards have a closed knight's tour ^[5].

Dissection puzzles grew out of a theorem proved in the 19th century: any polygon can be cut into a finite number of pieces that can be moved and turned to form any other polygon of the same area ^[4]. The theorem says nothing about how few pieces are needed, and that question of economy became the heart of dissection puzzles ^[4]. In most cases, the smallest number of pieces for a given pair of shapes is still unknown ^[4].

The 70 square was settled by treating it as a constraint-satisfaction problem, where every rule limits the choices that remain and a computer tries placements one after another ^[6].

How is a puzzle different from a plain exercise?

An exercise asks you to apply a method you already know. A puzzle hides its method. In every example above, the rules are short and easy to state, yet the route to the goal is not obvious, and finding it is the whole point.

That is also why puzzles reward reasoning more than speed. A puzzle that is both good and clear, in the sense of the 2020 paper, has a single answer that can be reached from its statements alone ^[3], so a patient solver with no special knowledge can always get there.

Try it yourself

Here are four quick ways to test the definition on puzzles you already know.

• Pick any puzzle and name its starting position, its rules and its goal. If one of them is missing or vague, the puzzle will probably feel unfair.
• Check whether it has exactly one answer. If you can find a second answer that follows every rule, the puzzle is not good in the strict sense.
• Look for its dead ends. The tempting moves that lead nowhere are usually what makes it fun.
• Try to change one rule and see what happens. Allow a peg to jump two pegs at once, or let a tile move diagonally, and notice how quickly a hard puzzle can become trivial or impossible.

The riddles below each have one starting picture, a short rule and a single answer.