Tangram square with 3 pieces

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

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

Tangram square using 1 piece - the square

2 pieces

Tangram square using 2 pieces. Either 2 large triangles. Or 2 small triangles.

3 pieces

Tangram square using 3 pieces. 1 medium triangle and 2 small triangles.

4 pieces

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

5 pieces

Tangram square using 5 pieces. 1 medium triangle, 1 square, 1 parallelogram and 2 small triangles.

6 pieces

No solution is possible. See below for an explanation.

7 pieces

Tangram square using 7 pieces. 2 large triangles, 1 medium triangle, 1 square, 1 parallelogram and 2 small triangles.

For those of you who like their data summarised:

Number of piecesNumber of solutions
11
22
31
45
52
60
72

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?

All 13 possible Tangram squares.

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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4 in a row games: rediscovering Connect Four and its Forgotten Relatives

Connect Four is a game I remember playing a lot as a child, and I use it now at my Curious Minds Club. It’s an excellent introduction to abstract strategy games for children, and if you have not yet played it with your child then do have a go. The sound as you open the under-bar and the counters clatter onto the table will transport you back in time. Supermarkets often sell a generic version with a name like ‘4 in a Row’ or ‘Line Up Four’. The game has been spun off in several ways over the years: a giant version for the garden or school playground; a version which involves bounding discs off a table; a card game for up to four players.

When it was introduced in the form you recognise today by Milton Bradley in 1974 it kind of swept away any competition and is the only 4 in a row game most people can name. A search for ‘4 in a row game’ on Amazon only turns up Connect Four and its many generic versions. This overshadows the fact that there are some other good ones out there.

Flatten the Grid: Play On Paper

Before we get to those, first get a different take on the game by playing it on a horizontal surface instead of the usual vertical game grid. Print this game board on the thickest paper you can fit in your printer. You need 21 counters in one colour, and 21 in a different colour. You can use many things as counters e.g. buttons, beads, shells, coins. Slide your counter along as indicated by the arrows until you reach either the bold ‘baseline’ or another counter.

All the normal ways of making your 4 in a row apply. Do you find it easier to spot the 4 in a row your opponent is about to make so you can block them? Try facing the board in different directions: how does it affect the way you play the game? A winning strategy is to make a 3 in a row which is open at both ends: on their next move your opponent can only block one end, leaving you able to win on your next move. Do you find it easier to apply this winning strategy when playing the game this horizontal way?

Step Into the Third Dimension

Whether you play Connect Four in this horizontal way or the usual vertical way it is a two-dimensional game. One interesting twist came when a third dimension was added to the game. A company called Funtastic brought out a game called Score Four in 1967. Another company called Lakeside took it on in the 1970s and it became more popular. Second hand copies can be bought online. The normal ways of making your 4 in a row apply (vertical, horizontal or diagonal) but with the addition of building rows on or across different levels. There are more ways to win, and more ways to lose!

Other three-dimensional games:

Eternas (2011) with a board in the shape of a circle;

Helix (1974), in which the board has overlapping arcs;

Quadrago (2007) with four middle bars that rotate to add complexity to the board.

There are more to be discovered if your interest has been piqued.

Think Outside the Grid: Brainline

Another obscure, vintage 4 in a row game is Brainline by Palitoy (probably released in the 1970s). The game board is made up of hexagons. Each player has four pegs which start the game on a specific hex. Players take turns to move any one of their pegs any distance along a straight line in any direction but may not jump any other peg. The winner is the first to get all four pegs of their colour in a straight line. The little twist I like is that the pegs do not have to be adjacent, but there must be no opponent pegs between them. Many times, you feel you are about to win, before your opponent thwarts you by moving their peg just where you don’t want it.

Circular Geometry

One final game to mention is called Circular Tic Tac Toe (or Circular Noughts and Crosses). A circular board is divided into 32 segments. There are four different ways to make a 4 in a row: a line from the central segment to the outer segment; a spiral to the left getting further away from the centre; a spiral to the right getting further away from the centre; and four adjacent segments of the same colour. The first to play has an advantage, as happens with many abstract strategy games. The Pie Rule can be used to level things up.

I was unable to find a commercial version of Circular Tic Tac Toe so I drew the board on a piece of MDF, coloured in the segments and took some counters from another game. Here is what it looked like:

Circular Tic Tac Toe before the game begins

Here is an example of a win: White has won with a spiral to the right getting further away from the centre.

White's winning spiral in Circular Tic Tac Toe

Taking it Further

If you have five minutes to spare, I recommend the Numberphile You Tube video on Connect Four. You will learn that it is a mathematically ‘solved’ game and be amazed by how many possible configurations of the grid there are. 4.5 trillion is a mind-boggling number.

Whether you stick to the classic red and yellow gravity-obeying counters or branch out into circular boards or three dimensions, I hope this inspires you to look at 4 in a row games with fresh eyes.

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How I got the yips solving my Rubik’s Cube, and how I fixed myself

I taught myself to solve a Rubik’s Cube over two weekends in early June 2026. All was going well until a few weeks later.

What motivated me to embark on a task I knew was going to be difficult? I had two motivations.

It had been on my mind for years as something I wanted to do, but I kept putting it off as it seemed too high a mountain to climb on my own. The Rubik’s Cube was massive in the 1980s, the decade of my childhood. While I can’t remember owning one, tv programmes regularly featured someone setting a new record, or solving it in a novel way.

Secondly, a Y5 boy attended my after-school Curious Minds Club at the library with his mum. Before we started, she gave him a parcel which must have arrived while he was at school. He took great delight in unwrapping his speedcube and telling us about it. I thought, I can’t have a 10 year old boy who knows something I don’t. That spurred me on to have a go. I had some generic cubes in my many boxes of games. I retrieved them from the attic and made a start.

Six Generic Rubik's Cubes

I quickly realised I could not learn by following illustrated instructions. Thank goodness for YouTube. I discovered Easiest Solve, and was hooked by his emphasis on ease not speed. He uses stories and images, not notations and algorithms, and goes nice and slow with lots of repetition. Just what my brain needed.

I worked through all eight of his beginner videos. Over two weekends, making my own notes for the hardest steps, and with hours of practice …. I could do it. All by myself. Consistently and correctly. I could even do the final step with my eyes closed. A lifetime’s ambition ticked off. I was mildly proud of my best time of 2m, 28s, although not tempted to learn speedcubing to bring it down below one minute. It was enough that I could solve any scrambled cube at will.

All was going well until mid-July. Since June, I hadn’t been picking up a cube as often. One afternoon, I mixed up a cube. Everything went well until step seven. I just couldn’t solve it. I returned myself to the start of step seven many times and every time it went wrong. I tried a different cube, as if that would change my luck. Of course it made no difference.

I was convinced I was doing it right. I started to panic, thinking that I would never solve the cube ever again. If you are a catastrophiser you will be able to relate to my state of mind. You might agree that catastrophising is a bad thing to be good at.

I watched Easiest Solve’s step seven again, and it was obvious where I was going wrong: at the second move, I was going counter-clockwise, when I should have gone clockwise. Putting myself right, I was able to solve the cube. My sense of relief was mixed with dismay at how quickly I had forgotten something I thought I had learnt. My muscle memory was much shallower than I had imagined.

In golf and other sports, professionals can suffer from the yips. They suddenly lose the ability to perform a skill they have mastered, due to a mental block or involuntary muscular spasm. My problem was one of forgetting something I had recently learnt. I will call it learning decay before reaching true mastery. It did definitely feel like I was blocked and full of anxiety. Perhaps the yips comes closer than another word.

I added ‘solve cube’ to my daily paper to-do list. I make myself sit down every day, scramble a cube, and solve it. I have my notes for step seven next to me. I’m now at the point where I don’t need to check them, but I still feel the need to keep them close.

Perhaps by making the Rubik’s Cube a daily task I am turning it into something prosaic, like remembering to put the tablet inside the dishwasher. But as I went to all the effort of learning it, I don’t want to lack that knowledge at my fingertips.

The technique of writing the digits of Pi down the left-hand side of my daily to-do list is how I have memorised the first 48 digits. I group them in threes and list them out: 141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375. Writing them every day helps embed the memory.

What an incredible invention the Rubik’s Cube is: it’s more than fifty years old and there are still plenty of children (and adults like me) who can’t resist the urge to pick one up and test themselves against the 43 quintillion possible configurations. But when you reach the top of the mountain you will stumble back down if you don’t keep practicing your new skills.

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