Impossible tangram variations online

The impossible tangram variations

Each puzzle here is a dissection of its own, played with pieces nothing else on the site uses, and makes only the one shape they are cut from. The Stomachion is the oldest of the tangram's cousins on record: fourteen pieces cut from a square, ten of them triangles, which Archimedes wrote about in the third century BC. The pentominoes are the twelve shapes five squares make edge to edge, named by Solomon Golomb in 1953, and fill a six by ten rectangle in 2,339 different ways.

Neither set has a piece cut at 45°, so here a piece turns a quarter at a time. Most of the pieces have a mirror image that no turn will reach, so expect to flip them over too.

Both are rated impossible, past the easy, medium and hard the rest of the library is measured on. Each is one plain shape that gives away nothing but its own sides, filled with more pieces than a tangram has, so there is no corner to start from and no silhouette to read. Both can be done: each is drawn from the pieces as they lie.

How to play

Not sure where to start? How to play has the rules and the controls, and walks through a whole figure one piece at a time. If you would rather learn with the pieces in your hands, the interactive tutorial solves the square with you.

How difficulty is rated

There are four difficulties, each with the tag it wears on the shelf and over the board:

• Difficulty: easyThe outline gives most of the pieces away.
• Difficulty: mediumSome of the pieces have to be worked out.
• Difficulty: hardThe outline gives almost nothing away.
• Difficulty: impossibleOnly the variations, with pieces of their own and one plain shape to fill.

Every tangram puzzle is rated easy, medium or hard, and the rating is measured from the shape rather than guessed. What makes a tangram hard is how little its outline gives away: a piece with its edges on the outside of the figure shows where it goes, while a piece hemmed in by other pieces has to be worked out.

To rate a puzzle, we add up the length of piece edge hidden inside the figure, where one piece meets another and the silhouette shows nothing. The more of it there is, the harder the puzzle. That is why a figure with parts sticking out is usually easy, and a plain square, which looks the simplest of all, is among the hardest. We checked the measure against how long people take over their first solve of each puzzle, and it holds.