DIY · Card 81 · Mathematics · Shapes
One square of card, five straight lines, seven pieces. Rearrange them into cats, boats, swans and running people, and every one has exactly the same area as the square you started with.
01The brief
Seven pieces, endless pictures, one area.
7
pieces cut from one square by five straight lines
16
small triangles’ worth of area in every picture
13
shapes without dents the pieces can make, and no more
The tangram is the simplest puzzle there is to make and one of the hardest to put down. Five cuts across a square give seven flat pieces; the challenge is to rearrange them, all seven, into a picture.
Along the way you meet fractions, areas, angles and the square root of two, all without a single sum. It goes well with the missing square puzzle and the spiral of Theodorus.
02Kit
A square of card, a ruler and scissors.
Tick off what you have
0 of 4 ready
03Method
Five lines, in order, then cut, colour and play.
Step 1 of 8 · Minute 0
You need a square of thick card about 16 cm across, a ruler, a sharp pencil, scissors and some coloured pens or paints.
The pieces don’t fit back into a square.
Why
A line was drawn to the wrong point.
Fix
Check every line starts and ends on a corner, a midpoint, or the first diagonal. Measuring in 4 cm steps helps.
Gaps appear between pieces.
Why
The cuts wandered off the lines.
Fix
Cut slowly, or score thick card with a craft knife against a ruler before snapping it.
A picture won’t come out.
Why
The parallelogram is the wrong way round.
Fix
Flip it over. It is the only piece that changes when turned face down.
04Pieces and areas
Every angle in a tangram is 45°, 90° or 135°, so pieces turned in 45° steps always line up. Here is a table to play on:
Try to make, using all seven pieces:
Drag a piece to move it and tap it to turn it by 45°. Corners click together when they come close. Every shape you make has the same area as the square, 16 small triangles’ worth, however different it looks.
Now look at what each piece is worth:
Of the square
1⁄4
Small triangles
4
Edges
4 · 2√2 · 2√2
Count in small triangles and the whole square is 16: two large triangles of 4, and the medium triangle, the square and the parallelogram of 2 each, and two small triangles of 1. So the square and the parallelogram have exactly the same area as the medium triangle, though they look nothing alike. Every edge is √2, 2, 2√2 or 4 units long, which is why the pieces fit together in so many ways.
05The big idea
Area
is conserved.
Cut a shape into pieces and rearrange them, and the area never changes, however different the new outline looks. That is one of the oldest ideas in geometry: Euclid used it to prove Pythagoras’ theorem.
It is also why the missing-square puzzle is a trick and the tangram is not: if the pieces fit exactly, nothing can be gained or lost. A cat and a swan made from one tangram contain exactly the same amount of card.
06The math
1 · The areas
4 + 4 + 2 + 2 + 2 + 1 + 1 = 16
In small triangles; the whole square is 16.
2 · The lengths
√2, 2, 2√2, 4
Each is √2 times the one before.
3 · The angles
45°, 90°, 135°
Every corner is one of these.
With a 16 cm square, each unit is 4 cm. A large triangle has legs of 2√2 × 4 ≈ 11.3 cm and a long side of 16 cm; its area is 64 cm², a quarter of the square’s 256 cm². The small triangle is 16 cm², a sixteenth.
Each triangle is half of a square on its long side, so its legs are √2 times shorter than that side. Going from small to medium to large triangle, every length grows by √2 and every area doubles: 1, 2, 4.
07Real life
From a 19th-century craze to classrooms and packing robots.
A Victorian craze
Tangram books reached England in 1817 and within two years were being printed across Europe and in the United States. Cartoonists drew people so absorbed they forgot to eat.
Fractions you can hold
Teachers use tangrams to show that a quarter, an eighth and a sixteenth can look completely different yet still fit together into one whole.
Dissection puzzles
The tangram belongs to a family of puzzles that cut one shape into pieces that make another, like Archimedes’ Stomachion, a 14-piece square puzzle from over 2,000 years ago.
Paradoxes
Some tangram pictures look identical except that one has a small piece missing, yet both use all seven tans. The famous ‘two monks’ puzzle shows two near-twin figures, one with a foot and one without.
Pixel art and icons
Designers who make logos from simple geometric pieces use the tangram’s trick: a few fixed shapes, turned and combined, can suggest almost anything.
Robots and packing
Computer scientists use tangram-style problems to test programs that pack and arrange parts, from loading lorries to cutting fabric with as little waste as possible.
1813
The earliest known tangram book was printed in China in 1813, though sets survive from a few years earlier. In Chinese the puzzle is qiqiaoban, the ‘seven boards of skill’.
Ships brought sets and books to the West, and in 1817 a London publisher issued The Fashionable Chinese Puzzle. Within two years tangram books had appeared in France, Germany, Italy, Denmark, the Netherlands and the United States: one of the first worldwide puzzle crazes, a century before the crossword.
08Is and isn’t
The tangram is thousands of years old.
That story was invented in 1903 by the puzzle writer Sam Loyd. The earliest known tangram book dates from 1813.
Bigger-looking shapes use more card.
Every picture made from all seven pieces has exactly the same area as the original square.
You can make any outline you like.
Many outlines are impossible. Only 13 shapes without dents can be made at all.
The square and the parallelogram are different sizes.
They have exactly the same area, each two small triangles, the same as the medium triangle.
09Go further
A
The 13 convex shapes
Try to find all 13 shapes with no dents. The square, triangle and rectangle are the easy ones.
B
Two tangrams
Make a second set and combine all 14 pieces. How many new shapes can you find?
C
A tangram alphabet
Make every letter from A to Z using all seven pieces each time. Which letters are hardest?
D
Measure the angles
List every angle in every piece. Why are they all 45°, 90° or 135°?
E
Design a puzzle
Make a picture, trace its outline only, and challenge a friend to fill it.
10Questions
Draw one diagonal of a square, join the midpoints of two neighbouring edges, draw half of the other diagonal up to that line, then add two short lines to separate a small square, a small triangle and a parallelogram. Cut along the five lines to get seven pieces.
Two large triangles, one medium triangle, two small triangles, one square and one parallelogram. Measured in small triangles, their areas are 4, 4, 2, 1, 1, 2 and 2: sixteen in all.
From China, where it was known as the ‘seven boards of skill’. The earliest known tangram book was published there in 1813, and around 1817 the puzzle swept through Europe and America as a craze.
Exactly 13. In 1942 the mathematicians Fu Traing Wang and Chuan-Chih Hsiung proved that the seven pieces can form only 13 shapes with no dents, including the square, a triangle, a rectangle and a parallelogram.
They make area, fractions, angles and symmetry something you can hold. Every picture has the same area as the square, every angle is a multiple of 45°, and every edge is one of four lengths related by √2.
Safety · Completely safe. If you cut thick card with a craft knife, a grown-up should do it on a cutting mat.
All DIY cards →Keep wandering
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