What Is a Mechanical Puzzle? Parts, Kinds and Why They Get Hard

A mechanical puzzle is a physical object that you solve by moving, fitting or rearranging its parts until they reach a goal arrangement. Nobody hands you clues or a rule sheet: the shape of the pieces and the frame that holds them decide which moves are possible, and your job is to find a sequence of those moves that ends at the goal.

That short definition covers trays of sliding tiles, boards of jumping pegs, sets of blocks that must be packed or matched, and shapes that are cut up and rebuilt. This article looks at what these objects have in common, how they differ from other puzzles, why a handful of plain parts can hide a very hard problem, and what a small sliding tray reveals when you count its positions.

What are the parts of a mechanical puzzle?

Every mechanical puzzle can be described with four parts.

• Pieces. What you handle: tiles, pegs, blocks, cut shapes, rings or rods.
• A frame. Whatever limits where the pieces can go: a tray, a board of holes, a box, or the outline of a target shape.
• Moves. The actions the object allows, such as a slide, a jump, a turn or a lift.
• A goal. The arrangement you want to reach, such as tiles in order, a single peg left, or every piece inside the box.

A sliding tile puzzle shows all four at once. The pieces are numbered tiles, the frame is a grid with one cell left empty, the only move is to slide a neighbouring tile into the empty cell, and the goal is to bring the tiles back to their starting order ^[1].

The word mechanical tells you where the rules live. In a logic grid or a number-place puzzle you read the rules and keep track of the state on paper. In a mechanical puzzle the state is the position of the parts in your hand, and the rules are built into the object itself.

What are the main kinds of mechanical puzzle?

This article sorts mechanical puzzles by the move each one asks of you, because the move decides how you think. The table compares six kinds.

The first four kinds have short, exact move rules, which makes them easy to state and to count. The last two are harder to pin down in a single rule, because their moves depend on the exact shape of each part.

Some puzzles mix kinds. A packing puzzle can also be an assembly puzzle if the pieces carry colours, and a dissection puzzle becomes a packing puzzle once the pieces are already cut and you only have to fit them into a frame.

Where do the rules of a mechanical puzzle come from?

In most puzzles the rules are words. In a mechanical puzzle they are walls, holes and edges. The tray of a sliding puzzle stops a tile from leaving the grid, and a tile can only move where there is a gap for it ^[1]. On a peg board, a jump is allowed only along the lines of holes, so the drilling of the board sets the moves ^[2].

A few rules are agreed rather than built in. In the theory behind dissection puzzles, pieces may be shifted and rotated to form the new shape ^[4], and some puzzles add a ban on flipping a piece over. The picture riddle below works with exactly that rule, so turning is allowed and flipping is not.

Built-in rules have a pleasing side effect. You cannot break them by accident, so any position you reach by honest moves is a legal one. The only way to create an illegal position is to take the object apart and reassemble it, and as the worked example below shows, that can leave a puzzle with no solution at all.

How is a mechanical puzzle different from other puzzles?

The difference lies in where the puzzle state is kept and how you change it.

A pencil-and-paper puzzle stores its state in marks on a grid, and you change it by writing. A riddle stores nothing: you change only your own understanding until the answer clicks. A mechanical puzzle stores its state in the physical position of its parts, and the only way to change that state is a legal move.

The material itself matters less than the moves. A sliding tray drawn on a screen follows the same rules as a wooden one, and a paper published on 10 July 2015 noted that the sliding puzzle had appeared in apps for mobile devices and as small puzzles inside larger computer games ^[1]. What makes both versions mechanical is the fixed set of moves, not the wood.

Picture riddles about machines sit right next door. In the riddle below you never touch a gear, yet you have to work out how each part forces the next, which is the core skill of mechanical puzzles.

Why do a few simple pieces make a hard puzzle?

Two things make mechanical puzzles hard, even when every move is easy to make.

Huge numbers of positions. Each arrangement of the parts is a position, and the count grows very fast. On the plus-shaped board with 33 holes, a paper published on 9 June 2017 found 3,626,632 different positions with 18 pegs that can still be solved down to a single peg in the centre, with rotated and mirror-image copies of a position stored only once ^[2].

Long routes. Even when you give up on the shortest route, the work grows fast. On a square sliding tray, a simple method that fixes the first row and column and then repeats on the smaller tray needs a number of moves that grows roughly with the cube of the tray width ^[1].

A third trap is less obvious. Some positions look almost solved but are impossible. That is the subject of the worked example.

How many positions does a small sliding tray have?

Take a tray two cells high and three cells wide. It holds five tiles, lettered A to E, and one gap. In the solved position the top row reads A, B, C and the bottom row reads D, E and the gap.

There are six cells, so the five tiles and the gap can be placed in six times five times four times three times two times one ways. That gives seven hundred and twenty arrangements in total.

Now start from the solved tray and try every possible slide, then every slide from each new position, and so on until nothing new appears. A short computer search that does exactly this reaches three hundred and sixty arrangements, which is exactly half. The other three hundred and sixty can never be reached by sliding, however long you try.

The search also finds the hardest reachable position. It needs twenty-one slides to solve, and only one arrangement in the whole tray is that far from the goal. Every count in this section comes from listing each reachable position with its distance from the goal, which anyone can repeat.

Why can half of the positions never be reached?

Here is a position that looks almost solved but sits in the unreachable half. It is the solved tray with tiles A and B swapped by hand.

The reason is a hidden balance. Colour the six cells like a chessboard. Every slide moves the gap onto a cell of the other colour, so the gap can only return to its home corner after an even number of slides. Each slide also swaps the gap with one tile, so returning home always takes an even number of swaps.

Swapping two tiles by hand, with the gap left at home, is a single swap: an odd number. No even run of swaps can produce it, so the swapped tray can never be solved. The same balance test decides every one of the seven hundred and twenty arrangements: a position is reachable exactly when its swap count and the gap's distance from home are both even or both odd.

This matters to anyone who takes a puzzle apart. If you reassemble a sliding tray and it refuses to finish with two tiles out of place, swap any two tiles by hand once more and it becomes solvable again.

What makes a mechanical puzzle a good one?

A good mechanical puzzle passes a few simple tests, and the two-by-three tray makes each of them concrete.

The goal is visible. You can see at a glance whether the tiles are in order, so there is never any doubt about whether you have finished.

The start can be reached. A maker who scrambles the tray by sliding always hands over a solvable puzzle. A maker who lifts the tiles out and drops them back at random is just as likely to hand over an impossible one as a solvable one, since exactly half of the arrangements are unreachable.

Short to state, long to solve. The tray's only rule fits in one sentence, yet its hardest position needs twenty-one slides, and you have to find them in the right order.

Hidden structure. The best puzzles reward an insight, not just patience. In one assembly puzzle described in a paper published on 25 September 2018, eight cubes from a coloured set build a double-size copy of another cube with matching inner faces, and each copy can be put together in two ways ^[3].

How should you start a mechanical puzzle?

The same few habits work on nearly every kind.

• List the moves. First find every move the object allows from the start. In the solved two-by-three tray, for example, the gap sits in a corner and only two slides are possible.
• Look for what never changes. The swap balance of a sliding tray is one example. A quantity that no move can change tells you at once which goals are impossible.
• Try working backwards. Playing peg solitaire backwards from the goal is the same as playing forwards from the reverse pattern, with every peg swapped for a hole ^[2].
• Accept steps away from the goal. In a sliding tray you may have to break up a finished row for a while so that the rest can move, and the same can happen with interlocking or disentanglement puzzles.

Which riddles use the same kind of thinking?

The picture riddles below each turn on a physical part: a valve, a gate that flips, or a rod you may move once. Every one of them gives a fixed scene, one rule to respect and exactly one correct answer.