Jigsaw Puzzle Strategy: Edges, Sorting, Clusters and the Last Pieces

A reliable jigsaw puzzle strategy is to work from what is certain towards what is not: turn every piece face up, pull out the straight-edged pieces and build the frame, then sort the rest by colour and by shape and grow small clusters until they join. The last pieces go in by shape rather than by picture, because by then each gap shows exactly which pattern of knobs and holes it needs.

Below you will find the method step by step, a table of sorting methods, a worked example that uses the frame to find the size of the grid, what the mathematics of clusters says about the hard part, and a practice puzzle with its answer.

In what order should you solve a jigsaw puzzle?

Work in this order. Each step makes the next one smaller.

1. Turn every piece picture side up. Spread them on a clear surface so none overlap. It takes a while, but every later step depends on seeing each picture.

1. Pull out the edge pieces. A piece with one straight side belongs on the border. A piece with two straight sides at a right angle is a corner.

1. Build the frame. Corners first, then the four sides. Count the border pieces as you go.

1. Sort the interior pieces. Group them by colour first, then split big colour groups by pattern or texture.

1. Grow clusters. Pick a strong feature, such as a straight line, a block of lettering or a sharp change of colour, and build it into a small finished patch.

1. Place the clusters. Use the picture on the lid and the frame to move each patch to roughly the right spot, then join it to its neighbours.

1. Sort what is left by shape. When the colours run out, the knobs and holes take over.

1. Fill the last gaps one at a time. Read each gap, choose a piece with the matching shape and check that the picture runs on.

How should you sort the pieces?

Sorting turns one huge search into many small ones. The table compares the common ways to sort.

Use small trays or shallow boxes for the groups. Keep a separate group for pieces you cannot place yet, and revisit it whenever a cluster grows.

How many shapes can a jigsaw piece have?

On a puzzle cut as a grid, each side of a piece is flat, a knob that sticks out, or a hole that takes a knob. Inside pieces have four sides, each a knob or a hole. You can turn a piece in your hand, but you can never flip it, because the picture must face up.

Because turning is allowed, an inside piece comes in just six basic shapes, as you can check by listing every pattern of knobs and holes:

An edge piece is different. Its straight side fixes which way up it goes, so its other three sides can form eight patterns. A corner has two straight sides, which leaves four patterns.

This gives a fast shape sort. Put the inside pieces into six piles, and when a gap needs two knobs on opposite sides, you only search one pile. Puzzles with irregular cuts break this rule, so treat it as a tool for grid-cut puzzles.

How do you build the frame?

Start with the four corners, then lay out each side. Put each straight edge against the table edge or a ruler so the sides stay true.

The frame also gives you a free check. If the puzzle is laid out in rows and columns, the border holds twice the number of rows plus twice the number of columns, minus four, because each corner sits on two sides at once. To get the inside count, take two off the number of rows, take two off the number of columns, and multiply the two results.

If your border pile does not match any grid, something is off. An edge piece may be mixed in with the inside pieces, a piece may be missing, or the puzzle may not be cut as a plain grid. Check the inside pile first, before you start the inside.

How does the frame tell you the size of the grid?

Here is a worked example. The lid says three hundred pieces. After the first sort, your edge pile holds seventy pieces, corners included. How many pieces long is each side?

Step 1: list the likely grids. Three hundred can be laid out in several ways. Here are the ones with at least ten rows.

Step 2: turn the border count into a sum. Add four to the seventy border pieces and halve it. That gives thirty-seven, which is the number of rows plus the number of columns.

Step 3: find the pair. You need two numbers that add to thirty-seven and multiply to three hundred. Twelve and twenty-five work, since twelve twenty-fives make three hundred.

Step 4: use the answer. Each long side holds twenty-five pieces and each short side twelve, corners included. The inside holds ten rows of twenty-three, which is two hundred and thirty pieces.

Step 5: check as you build. When a long side reaches twenty-five pieces, it is complete. If a long side stops at twenty-four with no edge pieces left, one is missing, and you know to look in the inside pile.

You can confirm by listing every way to lay out three hundred pieces that no other grid has exactly seventy border pieces, so the answer is unique.

How do you build clusters inside the frame?

Pick the most distinctive part of the picture first. Bright objects, straight lines, lettering and sharp borders between colours give pieces you can match on sight.

Before you join two pieces, run two checks. The shapes must lock with no gap and no lifting, and the picture must carry on across the join. A shape match with a broken picture is wrong, and so is a picture match that needs pressing.

Build each cluster as a solid patch, not a long thin line. A patch has more open sides, so more loose pieces can join it. When two clusters reach each other, join them and keep going.

What does the mathematics of clusters say about the hard part?

A paper published on 16 June 2017 studied jigsaw percolation, a model of a group of solvers putting a puzzle together ^[1]. Every member holds a piece, and the same set of pieces carries two kinds of link: one records who knows whom, the other records which pieces fit ^[1].

In the model, two clusters merge only when they are joined by a link of each kind, and the puzzle counts as solved when everything ends in a single cluster ^[1]. The paper looked at the case where both kinds of link are laid down at random ^[1].

For that case the paper pins down, up to a constant factor, the point where solving switches from very unlikely to very likely ^[1].

Its result assumes that each link density on its own is at least a large constant times the logarithm of n divided by n, where n is the number of pieces, and it describes what happens as n grows ^[1]. Under that condition the switch depends on the product of the two link densities, and it sits near one divided by the product of n and the logarithm of n ^[1].

The proof also shows where the difficulty lies ^[1]. Any successful run needs a first group of about log n pieces that holds together through its own links alone, and the paper treats the appearance of such a group as the main bottleneck ^[1].

In the last stage of the proof, again using that condition on the link densities, once a group holding a fixed share of all pieces exists, every remaining piece is very likely joined to it by both kinds of link, so the rest follows ^[1].

This next point is our own comparison, not part of the model above, since that model describes a group sharing pieces rather than one person at a table. It still suggests a sensible habit: put your early effort into one solid starting section, because every piece you add to it gives the loose pieces more places to join.

How many ways can plain two-square tiles fill a rectangle?

Plain pieces with no knobs can fill a space in many ways. Take plain tiles, each made of two squares side by side. Listing every case gives the number of ways such tiles can fill small rectangles:

Knobs and holes rule out many of these arrangements, because touching sides must match. The six shape classes are coarse, but the exact size and position of each knob differ from piece to piece. So use the shape piles to narrow the search, and let the lock test and the picture decide each fit.

The counts above hide a pattern. A paper published on 16 May 2005 proved that whenever a rectangle has sides of k times n and k plus one times n, its number of fillings with two-square tiles is odd ^[2]. Every rectangle in the table has that form, and every count in it is odd.

The same paper proved that the number of such fillings of a region is odd or even exactly as the count for a certain smaller part of that region is ^[2]. With that result it settled a question about squares with a square hole cut from the centre, posed in 1997 ^[2].

Such a square has side 2n with a centred square hole of side 2m, and the paper proved its count of fillings with two-square tiles is 2 to the power n minus m, times the square of an odd number ^[2]. Our own computer count agrees for a six by six square with a two by two hole: 1,444 fillings, which is four times nineteen squared, as that result predicts ^[2].

How do you place the last pieces?

1. Count the gaps and the loose pieces. They should match. If they do not, check the floor and the sorting trays before assuming a piece is lost.

1. Read each gap as a shape. Note which sides need a knob and which need a hole, and which way the pattern runs.

1. Sort the loose pieces into the six shape piles. Each gap now points to one pile.

1. Turn, never flip. Try a candidate in all four turns. If none locks cleanly, it belongs somewhere else.

1. Check the picture last. A colour line or texture should run straight through the new piece.

Which mistakes should you avoid?

• Forcing a piece. A near fit that needs pressing is a warning sign, and a wrong piece blocks the right one later.

• Sorting too finely. Many tiny colour piles can cost more time than they save. Start with a few broad groups.

• Ignoring the frame count. A short side is the quickest sign of a stray edge piece.

• Building long thin lines. Solid patches give more places to join.

• Trusting the lid too much. Cropped or enlarged lid pictures can put a feature in the wrong place. Let the frame decide positions.

Can you solve this practice puzzle?

A lid says five hundred pieces. After sorting, your edge pile holds eighty-six pieces, corners included. How many pieces long is each side of the frame, and how many pieces go inside it?

A hint: add four to the border count and halve it, then look for two numbers with that sum that multiply to five hundred.

Answer: the grid is twenty by twenty-five. Adding four to eighty-six gives ninety, and half of that is forty-five, which is twenty plus twenty-five. Twenty twenty-fives make five hundred. The inside holds eighteen rows of twenty-three, which is four hundred and fourteen pieces. Listing every layout of five hundred pieces confirms that no other has exactly eighty-six border pieces.

Ready to test your eye for shapes on a single gap? In this riddle the pieces may turn but never flip over, and you must decide which one fills the hole.