How to build a trinomial cube out of lego

One of The Troutbeck School pupils and I recreated our wooden Montessori trinomial cube set of (a + b + c)³ out of lego. Here are the basics: a = 5; b=4; c =2. Knowing this some of you will be set and can now build all 27 blocks.

lego trinomial cube dissected

lego trinomial cube completed

For the rest of us who spent some time piecing it all together this is how it progresses:

There are 3 layers to the cube.

The 5 high ‘a’ layer (4 layers of lego blocks) representing a³+2a²b+ab²+2a²c+2abc+ac²

lego trinomial cube a layer

The 4 high ‘b’ layer (3 layers of lego blocks) representing a²b+2ab²+2abc+b³+2b²c+bc²

lego trinomial cube b layer

The 2 high ‘c’ layer (2 layers of lego blocks)* representing a²c+2abc+2ac²+b²c+2bc²+c³

lego trinomial cube c layer

*The ‘c’ layer represents the biggest compromise as the height of a single brick is less than the width of 2 and the height of two blocks is greater than the width of 2. Here is a picture of a³, b³ and c³ to make that clearer as c³ looks the least like a cube!

lego trinomial cubes of a, b and c

Now follow your nose by building each of the 27 factors in the resultant equation. (As you got this far I know you can work that out). NB the 6 abc blocks will be in 3 pairs and are not quite identical but you will figure that out if you build layer by layer.

lego trinomial cube 3 pairs of ‘abc’ blocks

We nominally made ‘a’ out of red bricks, ‘b’ out of blue and ‘c’ out of yellow bricks but there are some compromises again, partly because of the limited brick colours available as the other children were making much more interesting models of cars, boats, space ships, lego friends etc.

Here we go block by block and layer by layer. Starting with the ‘a’ layer.

lego trinomial cube a layer

The ‘b’ layer.

lego trinomial cube b layer

The ‘c’ layer.

lego trinomial cube c layer

Next we build it up in time-lapse photography.

lego trinomial cube building a layer

lego trinomial cube building b layer

lego trinomial cube building c layer

lego trinomial cube building c layer – the completed cube

By the end you will realize that you don’t want to have to build it again and that the blocks love slipping around as you try to move them a layer at a time. Hence you will need to license the patented trinomial cube holder and lid shown below. The white gate hinges in the middle and can be removed and replaced as you use the cube.