I Solved Every Single Tangram Square (So You Don’t Have To)

On 14/09/202616/09/2026 By Games4LifeIn Recreational MathsLeave a comment

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. But not 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. 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.

1 piece

2 pieces

3 pieces

4 pieces

Tangram square using 4 pieces. 1 large triangle, 1 medium triangle and 2 small triangles.

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

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.

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.

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Does it bother you that 2 is the only even prime number?

On 03/09/202616/09/2026 By Games4LifeIn Recreational MathsLeave a comment

A number is prime if its only divisors are 1 and itself. A definition we are all familiar with. A fact we could teach a child once we think they are old enough to understand.

Given the definition, 2 is a prime number. Out of the infinite number of primes, 2 is the only one that is even.

But 2 never had a chance of not being prime. Contrast this with 3. We travel along the number line 1, 2, 3. We know that 1 and 3 are going to be divisors of 3 (as every number can be divided by itself and 1). There is a number – 2 – of which we can ask the question: is this a divisor of 3? The answer of course is ‘No’ so 3 is prime. But we got to ask the question. 3 had a chance not to be prime before we asked the question.

When you get to 4, you know that 1 and 4 are going to be divisors of 4. There are two candidates – 2 and 3 – and we ask if they are divisors of 4. Of course, 2 is a divisor and 3 is not a divisor. 4 is therefore not prime. It is the first composite number.

Going along the sequence of counting numbers there are always candidates for being divisors, but this never applied to 2: there is no whole number in there between 1 and 2 which 2 could have been divided by.

Don’t get me wrong: I am not of the opinion that 2 should not be a prime number. The prime numbers are the so called “building blocks” for all of the numbers: they allow us to construct all numbers from prime numbers in terms of multiplication. Without 2 being prime this would not be possible, and the “fundamental theorem of arithmetic” would fall apart. It is just that 2’s solitary evenness amongst a sea of odd primes feels untidily pattern-breaking. We say that all prime numbers are odd … apart from 2. We have to make an exception for 2. We have to explain that it is the only even prime number. That need to explain feels like a small imperfection in the abstract world of numbers, a world that stands in contrast to the eternally imperfect physical world we must inhabit every day.

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Further fun with the Archimedean Solids at the Curious Minds Club (St Thomas of Canterbury Primary School, 13 March 2020)

On 18/03/202016/09/2026 By Games4LifeIn Recreational MathsLeave a comment

This week at the Curious Minds Club we continued to build the 13 Archimedean Solids, with Polydron Frameworks and Magformers.

My Y2 girl who got half way through a Truncated Icosahedron two weeks ago (then had to miss last week’s session) was happy to finish it. She then got her first go at Magformers. She made several of the Platonic Solids from looking at a picture of the net, then made a lovely symmetrical pattern on her own initiative.

I asked my two Y1s to build a Truncated Cube in Polydron. Once they had got the alternation correct at the start (put a triangle on one edge of the octagon, miss one edge, add another triangle etc) they were able to bring the whole solid together. They needed a little help snapping it together at the end.

My Y5 boy completed the Icosidodecahedron in Polydon, having made it in Magformers last week. My Y4 girl made the Cuboctahedron in Polydron first, then in Magformers. She built the Rhombicuboctahedron in a Polydron net really quickly, then needed quite a lot of help bringing it together. With a few minutes left at the end I gave her the Tangram puzzle. She solved it in a few minutes with no help. My Y6 girl and Y6 boy attempted the Rhombicuboctahedron in Polydron. It didn’t go quite to plan. Bob was born instead.

Here are the photos:

Y2 girl. Truncated Icosahedron (you may know it as a football); three Platonic Solids (can you name them?)

Truncated Icosahedron and Platonic Solids 13 March Y2 girl