Tangram Puzzle Solutions: A Step-by-Step Method

To solve a tangram puzzle, place the two large triangles first, because together they cover half of the outline, then fit the medium triangle, the square and the parallelogram into the corners that are left, and finish with the two small triangles. If a gap refuses to close, flip the parallelogram over or move one large triangle to a different corner and try again.

This guide covers the seven pieces and the size rules that link them, a step-by-step method, two building blocks that do most of the work, and nine worked solutions, from a plain square to a house, an arrow, a boat and a candle. A practice puzzle and a live riddle wait at the end.

What are the seven tangram pieces?

A tangram set is a square cut into seven flat pieces: two large triangles, one medium triangle, two small triangles, one square and one parallelogram. Every target shape uses all seven pieces, laid flat, with no overlaps and no gaps.

All five triangles have the same shape. Each one has a right angle and two angles of 45 degrees, so each is half of a square cut along its diagonal. Only their sizes differ.

The easiest way to compare the pieces is to measure them in small triangles. Count the small triangle as one unit of area, and the whole set adds up to 16 units.

Three facts from this table do most of the work:

• A large triangle is worth two medium triangles or four small ones. The space one large triangle fills can often be filled by a group of smaller pieces instead.
• The medium triangle, the square and the parallelogram have the same area, two small triangles each, even though their shapes are different.
• Many edges match. The long side of a small triangle is as long as the short side of the medium triangle and the long side of the parallelogram. The long side of the medium triangle equals the short side of a large triangle. The side of the square equals the short side of a small triangle and the short side of the parallelogram. Matching edges are where pieces can meet flush.

Every corner of every piece is 45, 90 or 135 degrees. So every corner of a tangram outline is built from those angles, which is one of the best clues you have.

Why does a tangram always come out exact?

A tangram is a fixed-piece version of a dissection puzzle, in which one shape is cut up and the pieces are rearranged into another shape with the same area. According to the Wallace-Bolyai-Gerwien theorem, a result from the 19th century, two polygons with equal area can always be sliced so that a finite set of pieces from one rebuilds the other ^[1].

The catch is how many pieces you need. A paper published on 27 January 2023 looked closely at an equilateral triangle and a square of the same area ^[1]. Henry Dudeney's answer to his Haberdasher Puzzle turns the triangle into the square with four pieces, and the paper proved that no cut into three convex pieces can do the same job ^[1].

A tangram turns that question around. The cuts are fixed, and you search for outlines the seven pieces can fill. That is why every tangram target has exactly the area of the full set, and why counting area is always your first check.

How do you solve a tangram puzzle, step by step?

1. Check the area. A target uses all seven pieces, so it covers exactly 16 small triangles. If your own sketch is clearly bigger or smaller than the set laid out as a square, the sketch is wrong, not the pieces.
2. Place the two large triangles first. They cover half the area, so where they go decides almost everything else. Try them in the biggest open regions and against the longest straight edges of the outline.
3. Read the corners. A sharp 45-degree corner needs the tip of a triangle or the sharp corner of the parallelogram. A 90-degree corner can be a right angle of a triangle, a corner of the square, or two 45-degree tips side by side. A 135-degree corner is the wide corner of the parallelogram, or a right angle next to a 45-degree tip.
4. Work from the border inward. Fit pieces against the outline before you fill the middle. Long border edges suit the large triangles, and short edges suit the small pieces.
5. Place the medium triangle, the square and the parallelogram next. Each covers two small triangles, so treat them as the middle-sized pieces they are.
6. Finish with the two small triangles. They are the most flexible pieces and fill the last notches.
7. If a gap is the right size but the wrong way round, flip the parallelogram. Of the seven pieces, only the parallelogram changes when you turn it over: its mirror image cannot be reached by turning it on the table. The triangles and the square look the same after a flip, so flipping them never helps.
8. Back up a little, not all the way. When you get stuck, lift the last two or three pieces and try a new spot for them. Start over only if the large triangles themselves are in the wrong place.

The border-first habit in step 4 is also how one research team attacked a much larger fitting puzzle ^[2]. A paper published on 25 September 2018 studied whether the squares of side 1 to 24 could exactly fill a square of side 70, and it worked by first listing which squares could sit along each edge ^[2]. It showed that in such a filling, the squares of side 1 to 5 could never touch the border at all ^[2].

Small pieces belong in the gaps, and big pieces settle the edges.

Area alone does not guarantee a fit, though. The squares of side 1 to 24 have exactly the same total area as a square of side 70, yet they cannot fill it, and the best possible attempt leaves out the square of side 7 ^[2]. So treat a matching area as a first check, not a promise.

Which two building blocks solve most shapes?

Most tangram solutions are easier to see if you stop thinking of seven pieces and start thinking of two blocks of equal size.

Block A: the two large triangles. Together they cover 8 small triangles. Put their long sides together and they make a square. Put two short sides together and they make a bigger right triangle whose long side is 4 units. Join two short sides with the triangles pointing opposite ways and they make a parallelogram.

Block B: the other five pieces. The medium triangle, the two small triangles, the square and the parallelogram also cover 8 small triangles, the same as Block A. They can copy Block A's two most useful shapes:

• The five-piece triangle. Lay the medium triangle with its point down and its long side up. On that long side, set the square over the left half and the parallelogram over the right half, leaning out to the right. One small triangle fills the far left corner beside the square, and the other fills the notch between the square and the parallelogram. The result is a triangle with a long side of 4 units, exactly like the one Block A makes.
• The five-piece square. Lay the medium triangle along the bottom with its point at the centre. Put a small triangle in the gap at the bottom right and the square in the top-right quarter. Run the parallelogram up the left side, from the bottom-left corner to the middle of the top edge, and drop the other small triangle into the top-left corner.

Once you can build these two blocks without thinking, most geometric targets reduce to one question: where do the two blocks go?

How do the pieces make basic geometric shapes?

Every arrangement below was checked by a small program that confirms the pieces have the right sizes, never overlap, and fill the outline exactly. Lengths are in the same units as the table, where a small triangle has short sides of 1.

How do you make the square?

Put Block A together as a triangle, point up. Build the five-piece triangle, point down, and push the two long sides together. The result is a square of side about 2.83 standing on one corner. Turn it a quarter of the way round and it sits flat.

How do you make the rectangle?

The target is 4 units wide and 2 units tall. Put the five-piece triangle in the middle with its point touching the bottom edge and its long side forming the top edge. A large triangle then fills each bottom corner, with its right angle in the corner of the rectangle.

How do you make the large triangle?

The target is a right triangle with two short sides of 4. Build the five-piece square and put it in the right-angle corner. One large triangle sits on top of the square, and the other sits to its right. Each one has its right angle touching the square.

How do you make the parallelogram?

The target has a base of 4, a height of 2, and sides that lean at 45 degrees. Put Block A together as a triangle, point up, on the left. Put the five-piece triangle, point down, beside it on the right. The two triangles share a slanted edge, and their tips form the sharp corners.

How do you make the trapezoid?

The target is 6 units along the bottom, 2 units along the top, and 2 units tall. Put the five-piece square in the middle. A large triangle goes on each side, with its upright short side against the square and its slope forming the end of the trapezoid.

How do the pieces make simple objects?

Objects work the same way. Look for a part of the object that matches one block, then fit the other block to the rest.

How do you make the house?

The walls are Block A as a square. The roof is the five-piece triangle, point up, with its long side resting on top of the walls. The roof is 4 units wide and the walls only 2, so the roof hangs out by 1 unit on each side, like eaves.

How do you make the arrow?

The shaft is Block A as a square. The head is the five-piece triangle turned so its long side stands upright against the shaft and its point faces right. The head is 4 units tall and the shaft 2, so 1 unit of the head sticks out above and below.

How do you make the boat?

The sail is Block A as a triangle, point up. The hull is a trapezoid 5 units along the deck, 3 along the bottom and 1 unit deep, made from the other five pieces in a row: the medium triangle point down at the left end, then a small triangle, the square, the second small triangle, and the parallelogram forming the slanted right end. The sail sits on the deck flush with the left end.

How do you make the candle?

The holder is a low trapezoid 4 units wide at the bottom and 2 at the top. Build it from a small triangle at the left end, the square, the second small triangle, and the parallelogram at the right end. The body of the candle is Block A as a square standing on the holder. The flame is the medium triangle, point up, on top of the body.

What mistakes slow people down?

• Leaving the large triangles until last. They decide half the area. Placed late, they rarely fit.
• Forgetting to flip the parallelogram. If one gap is a mirror image of the parallelogram, turn the piece over.
• Thinking every right angle needs the square. Two 45-degree tips side by side make a right angle too, and so does the corner of any triangle.
• Building the target piece by piece instead of block by block. The five-piece triangle and square turn a seven-piece puzzle into a two-piece one.
• Starting from scratch after every failure. Move the last few pieces first. The large triangles are often already right.

Can you change the rectangle with one move?

Build the rectangle from the worked example: a large triangle in each bottom corner and the five-piece triangle in the middle. Now move exactly one piece so that the rectangle becomes a parallelogram whose top edge sits further to the left than its bottom edge.

Answer: Lift the large triangle from the left end and put it at the right end, with its right angle against the right angle of the other large triangle at the bottom.

The left end becomes a slope, the right end gains a slope that leans the same way, and the shape is now a parallelogram 4 units along the bottom and 2 units tall. A program that tried every one-piece move found only one other, moving the right large triangle to the left end, and that gives the mirror image, whose top edge sits further to the right.

Ready for a fitting puzzle you can play right now? Here the pieces may turn but never flip, so use step 7 in reverse: