Tangram Puzzles
Seven flat pieces, one outline, and no overlapping. Fit all seven tans into the silhouette exactly — no gaps, nothing sticking out. Every shape here is built by placing the pieces first and tracing round them, so it is always solvable, and the arrangement it came from is one tap away when you want it.
Also known as qiqiaoban — seven boards of skill — and as a dissection puzzle or seven-piece puzzle.
Related Practice
What Is A Tangram?
A tangram is a dissection puzzle: a square cut into seven flat pieces which are then rearranged to make other shapes. The pieces are called tans, and there are always the same seven — two large triangles, one medium triangle, two small triangles, a square and a parallelogram. You are shown an outline and asked to fill it with all seven, laid flat, without overlapping any of them.
What makes it a puzzle rather than a craft is that the outline hides the joins. A straight edge four units long might be one large triangle or two smaller pieces meeting; the silhouette will not tell you which, and only one reading will let the remaining pieces fit.
The Seven Tans
Every tan is built from the same right-angled isosceles triangle, which is why they combine so readily: every angle in the set is a multiple of 45°, and every edge is either a whole unit or a diagonal. Areas are given relative to the smallest triangle.
| Piece | How many | Area | Where it usually goes |
|---|---|---|---|
| Large triangle | 2 | 4 each | Half the puzzle between them. Place these first — along the longest straight edges. |
| Medium triangle | 1 | 2 | Bridges the large pieces to the small ones. Often fills a corner the large triangles cannot reach. |
| Square | 1 | 2 | Needs a right-angled gap with equal sides. The hardest to spot, so leave it late. |
| Parallelogram | 1 | 2 | The only piece that can be flipped over. If a shape will not close, this is usually why. |
| Small triangle | 2 | 1 each | Fill the last awkward gaps, and make up the thin arms two units wide. |
| Total | 7 | 16 | Exactly the area of the original 4×4 square, every time. |
That total is worth holding on to. Because the seven tans always come to the same area, every legitimate silhouette has that area too — so if you can fit six pieces and the seventh will not go, the gap you have left is the wrong shape, never the wrong size.
How To Solve A Tangram
There is a reliable order, and it is not the order most people use. The instinct is to start with the small pieces because they seem easier to place. That is exactly backwards — a small triangle fits almost anywhere, so placing one early tells you nothing and commits you to nothing useful.
- Place the two large triangles first. They are half the total area and they have the fewest legal positions, so they carry the most information. Look for the longest straight edges on the outline: a large triangle is nearly always sitting along one.
- Clear the thin parts next. An arm or spike two units wide can only be a small triangle or the parallelogram — nothing else is narrow enough. Narrow regions place themselves, so take the free information before tackling the open middle.
- Read the corners. Every corner of a tangram silhouette is 45°, 90°, 135° or 180°. A sharp 45° point has to be the tip of a triangle; it can never be the square. Each corner therefore rules pieces out before you try anything.
- Fit the square and parallelogram last. These two have the most possible placements and the least distinctive angles. They are what you drop into the gap that remains, not what you build around.
- If it will not close, move a large triangle. When the final two pieces refuse to fit, the error is almost always an early one. Shuffling small pieces cannot repair a wrong foundation, so undo a big one and try the other edge.
Why Every Shape Here Can Definitely Be Solved
This is the part that works differently from other tangram sites, and it is worth being precise about. Ordinarily a silhouette is drawn by a person, and then somebody has to work out whether the seven tans can actually fill it — many plausible-looking shapes cannot be filled at all, even when the area is right.
Here it runs the other way round. The seven tans are placed first, at random but never overlapping and always touching, and the puzzle you are given is simply the outline of the result. The solution therefore exists before the puzzle does. There is no bank of hand-drawn shapes to run out of, and no possibility of being handed something impossible.
The pieces sit on a lattice of quarter-square triangles, so a placement is a set of small triangles rather than a position with a tolerance. Overlap is then set intersection and the silhouette is set union — both exact, with no rounding anywhere in the puzzle logic. That is also what lets the board tell you immediately whether a tap is a legal placement.
Only Thirteen Tangrams Are Convex
One of the genuinely surprising facts about this puzzle: of all the shapes the seven tans can form, only thirteen are convex — meaning the outline has no dents, and a straight line between any two points inside stays inside. Fu Traing Wang and Chuan-Chin Hsiung proved that number in 1942, and the original square is one of the thirteen.
It is a useful thing to know while playing, not just a curiosity. Convex shapes are the hardest tangrams there are, because a dent in the outline is a clue and a convex shape has none. That is the reasoning behind the difficulty levels here: the more compact and dent-free a silhouette, the harder it is rated.
Where The Tangram Came From
The tangram is Chinese, and its name there is 七巧板 (qiqiaoban) — literally "seven boards of skill". The earliest known printed reference appears in a Chinese book dated 1813, though the idea may reach back further to the yanjitu, a Song dynasty set of banquet tables designed to be arranged into different configurations.
It arrived in Europe and America around 1815 and became a genuine craze within a few years, with puzzle books printed on both sides of the Atlantic. It has been in classrooms ever since, mostly for teaching area, congruence and symmetry — a child can discover that two small triangles make the square by simply putting them together, which is a more convincing demonstration than being told.
How This Compares With Other Tangram Sites
Written plainly, including where the others are ahead. If you want a specific picture to recreate — a cat, a boat, a running figure — a hand-drawn set is what you want, and this is not that.
| What you care about | Most tangram sites | PuzzleType |
|---|---|---|
| How many shapes | A hand-drawn set, typically 10–25. | 60 named families, and a fresh silhouette from every seed. |
| Guaranteed solvable | Yes, because a person checked each one. | Yes, structurally — the pieces are placed before the outline exists. |
| Seeing the answer | Often nothing, or a static picture. | The exact arrangement, drawn into the shape you were given. |
| Placing a piece | Drag and rotate handles. | Tap to place, on an exact lattice — no dragging, and it works with a thumb. |
| Recognisable pictures | Yes — the real advantage. | No. Generated shapes are abstract, not cats and boats. |
| Sharing one puzzle | A level number, if anything. | A permanent seeded link that rebuilds that exact silhouette. |
| Cost and sign-up | Usually free, often with ads. | Free, no account, nothing stored off your device. |
Tangram Puzzles Frequently Asked Questions
How many pieces does a tangram have?
Seven, called tans: two large triangles, one medium triangle, two small triangles, a square and a parallelogram. The Chinese name, qiqiaoban, translates as "seven boards of skill", so the count is built into what the puzzle is called.
What are the rules of tangram?
Use all seven tans, lay them flat, let every piece touch at least one other, and never overlap them. Pieces may be turned to any angle, and the parallelogram may be flipped over — it is the only tan that is not its own mirror image. A shape filled with six pieces, or with two pieces overlapping, is not a solution.
Is every tangram silhouette solvable?
Not in general. You can easily draw a shape with the correct area that no arrangement of the seven tans will fill. Every shape here is solvable, for a structural reason rather than because someone checked each one: the tans are placed first and the outline is traced around them, so the answer exists before the question does.
How many convex shapes can you make with a tangram?
Exactly thirteen, proven by Fu Traing Wang and Chuan-Chin Hsiung in 1942. A convex shape has no dents in its outline, and the original square is one of the thirteen. They are also the hardest tangrams to solve, because a dent is a clue and these have none.
Where did the tangram come from?
China. The earliest known printed reference is a Chinese book dated 1813, and the idea may go back further to the yanjitu, a Song dynasty arrangeable table set. It reached Europe and America around 1815 and became a craze almost immediately.
Are tangrams good for children?
They are a long-standing classroom tool for area, congruence and symmetry, and deservedly so — a child can discover that two small triangles make the square by doing it rather than being told. What they reliably build is fluency with the shapes and with turning them mentally. Wider claims about raising general intelligence are not well supported, and we are not going to make them. Start on the easy shapes, which have thin arms that place themselves.
Is there only one solution to a tangram?
Usually not. Many silhouettes can be filled more than one way, especially where two small triangles could be swapped for the square or the parallelogram in the same spot. Show the solution reveals the arrangement the shape was generated from — one valid answer, not necessarily the only one. If your solution fills the outline with all seven pieces and no overlaps, it is correct even if it differs from ours.
Can tangram pieces overlap?
No. A solution uses all seven pieces, laid flat, each touching at least one other, and none of them overlapping. An arrangement that covers the outline by stacking pieces is not a solution, which is why the puzzle is harder than simply matching the area.
What can you make with a tangram?
Anything the seven pieces will tile, which is a surprisingly wide range — animals, boats, people, letters and numbers, as well as plain geometric shapes. The classic sets published since the nineteenth century run to hundreds of silhouettes. The shapes here are generated instead, so the supply does not run out, though that does mean they are abstract rather than recognisable objects.
What do tangrams teach?
Geometry you can feel rather than memorise. Fitting the tans together is direct practice in area, congruence, symmetry and rotation — two small triangles make the medium one, and the same seven pieces make a square or a long thin rectangle without changing area. That is why they have been a school manipulative for well over a century.