Which side of the landmark?

Twenty-four positions, one figure

Hold a cube square-on and it has twenty-four different positions. Six faces can be the one pointing at you, and for each of those the cube can be spun four ways. Six fours are twenty-four. A solid built from cubes has the same twenty-four, and it is the same figure in every one.

Two things are worth being exact about. Those are the square-on positions, which are the ones the drawings use; a real object can be turned by any amount in between. And a solid with some symmetry looks identical in several of the twenty-four, so it may have fewer views that can be told apart.

The paper shows you one position and asks you to recognise another.

Everything else on that line is a genuinely different figure — not a trick of the drawing. That is worth saying, because children often suspect these questions of being unfair, and they are not.

At this level the turn is usually a tip onto another face, often followed by a spin.

The C.U.B.E. method

C — Count the cubes. Counting often rules out nothing at this level, because wrong options commonly use the same number of cubes. Do it anyway; it takes five seconds, and now and then it removes an option.

U — The unique bit. Choose a landmark and commit to it: the lone cube, the tallest column, the arm that sticks out sideways.

B — Both sides. What is joined to your landmark on each side of it, and how are those neighbours arranged around it?

E — Eliminate. Predict first, then read the options.

The B is where the marks are.

Telling a turn from a reflection

Here is the reliable test, and it is worth more than any amount of staring.

First, what does not work: saying where things sit on the page. The tall column is on the left is true of one drawing only. Give the solid a half turn and the column is on the right of the page, and nothing has been reflected.

What a turn does keep is how the parts are joined, and which way round they are arranged. So describe the solid using its own parts rather than the page.

Pick your landmark cube and three things joined to it that point three different ways and that you can tell apart: say a tall column, a long arm, and a single cube sticking out. Now say a sentence about them out loud: if the column is up and the arm points forwards, the single cube is on the right.

Turn the figure any way you like. Line yourself up again, column up and arm forwards, and the single cube is still on the right, wherever the drawing has put it on the page.

In the mirror image, it is on the left. Same cubes, same joins, opposite way round. That is the whole difference, and it is the test that still works when the drawing looks convincing.

The diagram was drawn for this page. Its solid has seven cubes: a centre cube, marked with a dot, with arms of one, two and three cubes. B is A given a quarter turn, and C is the mirror image. Make the sentence for this solid. Stand the three-cube arm up and point the two-cube arm straight ahead of you: in A and in B the single cube is on your left. In C it is on your right — the other way round, which is all the test needs.

Turning a figure in your head

Do not try to move the whole thing at once. A nine-cube figure is a lot to hold in mind, and it is easy to lose track of one part while moving another.

So move one thing. Decide which face is going to end up pointing at you. Work out where your landmark cube goes. Only then rebuild the rest around it.

And if an option survives your landmark check, check a second feature before committing — because one feature agreeing can be a coincidence. A second makes that much less likely, and the option is only right if the whole arrangement matches.

Where do pupils lose marks?

Choosing the mirror image. A very common way to lose the mark. It survives the cube count, the outline and the silhouette, and fails only on which way round the parts are arranged about the landmark.

Matching the outline instead of the arrangement. One option has the right cubes with one moved a place — it matches from a single viewpoint and stops matching the moment you check a second face.

Turning the whole figure at once.

Reading the options before predicting. Wrong options are built to look plausible, and one of them can talk you out of the right answer.

How can we practise this at home?

Build a small asymmetric figure from blocks — nine or so cubes, with one obvious sticking-out arm — and turn it slowly while your child says where the arm has gone before it arrives.

Then the important half: build the mirror image of it alongside, and let them try to turn one into the other. Failing to do that, physically, with their hands, teaches the mirror trap in a way no diagram manages.