Jigsaw Puzzles

The picture is cut up and the pieces are spread around a tray beside the empty frame. Pick a piece up — mouse or finger — and drop it close to where it belongs: within about a third of a piece-width of its own spot it snaps into the frame and locks there for good. Anywhere else it simply stays where you put it, on top of the pile.

Pieces also join off the frame. Drop a piece beside one of its true neighbours, lined up the way they meet in the picture, and the two lock together and move as one group from then on. Groups keep growing the same way, and a group dropped near its place snaps into the frame in one go. Nothing joins or locks unless it is correct, so a wrong drop never has to be undone.

• Edges hides every loose piece without a straight side, so the border can be sorted out of the pile.
• Scatter spreads the loose single pieces back across the tray, steering round any groups you have built.
• Ghost picture shows the finished picture faintly inside the frame, and the box lid in the corner of the tray shows it in full. Both are in the menu, and both are remembered.
• Keyboard: Left and Right walk through the loose pieces one by one, and Enter puts the chosen piece straight into its place — the whole puzzle can be finished without a mouse.

On a regular grid, as a traditional jigsaw is: the picture is always half as wide again as it is tall, and each piece count gets the grid whose pieces come out closest to square. At 24, 96 and 150 pieces the pieces are exactly square; at 12 and 48 they are an eighth wider than they are tall.

Every join between two neighbours is a knob on one piece and the matching socket on the other, and both are drawn from one curve — traced one way for the knob and the other way for the socket — so they fit exactly and fit nothing else. Each join gets its own knob position, size and lean, which is why two pieces with the same number of knobs rarely look alike. The outside edge of the picture is straight, and all four corners have two straight sides.

Because the border is the one part of a jigsaw you can find by shape alone. A piece with a straight side belongs somewhere on the rim of the picture, and a piece with two straight sides is one of exactly four corners, whatever the picture shows. The rest of the puzzle has to be found by colour and detail.

The share of border pieces falls as a puzzle grows. On the 12-piece puzzle 10 pieces have a straight side and only 2 do not; on the 48-piece it is exactly half; on the 150-piece it is 46 pieces, a little under a third. The count is always twice the width plus twice the height, less the four corners that would otherwise be counted twice — so the bigger the puzzle, the more of it the border leaves for you to work out from the picture.

They are painted on the spot by this page, from the puzzle's number, in one of 10 scenes: a mountain lake at sunset, the sun over the sea, a stained-glass rose window, hot-air balloons over farmland, a patchwork quilt, a starry night, an autumn forest, a city at dusk, a coral reef and tulip fields with a windmill. Each scene has 4 colour schemes, and the number also decides where the sun or the moon sits, where the horizon falls and which way round the picture faces.

Nothing is downloaded — there are no photographs at all — so a puzzle opens instantly and works offline, and puzzle #482137 is the same picture, cut the same way, on any device. Every scene is built out of many small things, from lit windows to stars, leaves and fish, so that a piece of sky or sea still carries something to recognise it by. Today's puzzle is the same picture for everyone, and the same cut for everyone who picks the same number of pieces.

The first known jigsaw puzzles were made in the 1760s by John Spilsbury, a London engraver and mapmaker, who pasted maps onto sheets of wood and cut them apart along the borders of the countries. Putting Europe back together was the geography lesson. Those early puzzles did not interlock — each piece simply lay beside the next — and they were an expensive, hand-cut luxury for well-off households.

If you enjoy putting a picture back together, Sliding Puzzle scrambles one inside its frame instead, Memory Match asks you to remember where each picture is, and Nonogram builds a picture square by square from clues.