I bought some wooden Tangram sets to use at my Curious Minds after school club. Inside a wooden frame fit seven differently-coloured pieces: two large triangles; one medium triangle; two small triangles; a square; and a parallelogram. The frame is a square, and all seven pieces fit together to fill the space with no gaps and no overlaps.
“No gaps and no overlaps” is the univeral requirement when tessellating shapes. A tangram is not a tessellation, but the two concepts are closely related in geometry. It seems plausible that the study of tessellations helped inspire the creation of geometric puzzles like the tangram. My blog on two dimensional tessellations has turned out to be one of my most popular posts.
Playing around with the seven tangram shapes, I quickly saw it was possible to take a subset of the seven pieces and make different sized squares. Despite my best efforts, I could not find a website, blog post or activity sheet which showed every possible square. There are a wealth of resources for animal, bird or flower outlines, plus plenty of exotic shapes. As much as I was tempted to solve Duck in a Hurry and the Contented Rabbit, there was nothing for the humble square. The challenge was obvious: to create my own post to fill this gap for others.
I spent a happy hour sitting on the floor figuring out every combination. Here they are, leaving out simple rotations and reflections (across an imaginary mirror line outside of the shape). I have also left out images where I could have swapped a small orange triangle for a yellow one, or exchanged the large blue triangle with the pink one: these are just colour swaps, not a new configuration of the pieces.
13 Square Solutions
1 piece
2 pieces
3 pieces
4 pieces
5 pieces
6 pieces
No solution is possible. See below for an explanation.
7 pieces
For those of you who like their data summarised:
| Number of pieces | Number of solutions |
| 1 | 1 |
| 2 | 2 |
| 3 | 1 |
| 4 | 5 |
| 5 | 2 |
| 6 | 0 |
| 7 | 2 |
| Total | 13 |
If you are tempted to spot a pattern between the number of pieces and number of solutions, I didn’t get anywhere either!
Why can’t you make a square with 6 pieces?
For each piece, we can express its size as a unit of area:
The small triangle = 1 unit
The square = 2 units
The parallelogram = 2 units
The medium triangle = 2 units
The large triangle = 4 units
For any square, the total area must be either 2, 4, 8 or 16 units. Did you notice these are the successive powers of 2? That’s not a coincedence. Every time you step up to the next allowed square size, you are exactly doubling the area of the previous square. This is scaling symmetry, a beautiful, repeating geometric pattern. I will explore this further in a future blog.
The biggest square we can make with all 7 pieces has a total area of 16 units (1 + 1 + 2 + 2 + 2 + 4 + 4).
The squares with 5 pieces have a total area of 8 units (1 + 1 + 2 + 2 + 2).
The squares with 4 pieces have a total area of 8 units (1 + 1 + 2 + 4). (Note that any of the 2-unit pieces work).
The squares with 3 pieces have a total area of 4 units (1 + 1 + 2).
The squares with 2 pieces have a total area of either 2 units (1 + 1) or 8 units (4 + 4).
The square with 1 piece has a total area of 2 units (2, or the square by itself).
For any combination of 6 pieces you choose, there is no way to make a square with either 2, 4, 8 or 16 units. Lets look at what happens if you try to make a square using exactly 6 pieces. You have to leave one piece behind, which ruins the required “power of 2” area:
Leave out a large triangle (4 units): Remaining area = 16 – 4 = 12 units.
Leave out a medium triangle, square or parallelogram (each 2 units): Remaining area = 16 – 2 = 14 units.
Leave out a small triangle (1 unit): Remaining area = 16 – 1 = 15 units.
Because areas of 12, 14, and 15 cannot geometrically form a square using the standard fixed angles of a tangram set, a 6-piece square is mathematically impossible! Play about with 6 pieces so you can see how they stubbornly refuse to fit into a square.
Is one of the 4-piece solutions a reflection of itself?
Someone asked me if the solution on the right is a reflection of the one on the left.
I stated above that I am leaving out simple rotations and reflections. To get from the left solution to the right solution requires a combination of reflecting and rotating, so I categorise them as two different solutions. You can either: reflect the left solution clockwise by 90 degrees, then reflect it; or, you can reflect the left solution, then rotate it anti-clockwise by 90 degrees. Either way, something interesting happens: the yellow and orange triangles swap places! Try it yourself and see if you can figure out what is going on.
Every possible Tangram square
For the sake of completeness, here they are in one image. Does it look like a giant tangram person? But which way is it facing?
When I took my tangram sets to my after school Curious Minds club they were very popular. I had three boys at that session. We got into a beautiful groove. For example, I told them to take the purple piece, the yellow piece and the orange piece and make a square bigger than the one before. They were really good at focussing on their own pieces and not sneaking a look to see someone else’s solution. As we progressed to 5 and 7 pieces they did need some assistance, but not as much as I expected. By this stage, they had developed a feel for how the pieces fit together. It was helpful that with 7 pieces they got to make the square inside the wooden frame, which naturally guided them to the total area of the solution.
Feel free to contact me if you think I successfully filled a gap in your tangram knowledge.
Ready for the next shape? Check out how I solved all 31 Tangram parallelograms.
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