Cube Counting

The type of question

You are presented with a stack of identical, small, 3D shapes. The drawing is from one, fixed position. These cubes (or cuboids) cannot be floating, and must be supported entirely along their bases.

The tricky part is that you can’t see all the blocks, so you need to picture how the visible blocks are being supported by other blocks in order to build a mental picture of how it fits together.

For cube stacks

If all of your shapes are cubes, count each stack individually. Make a note of how many are in each column. Your total is just the total of the blocks in those columns.

These are fairly simple, but always double check your answer, as it would be frustrating to get one of these wrong.

For trickier stacks

Remember, no cube is floating. If a block seems to be poking out from behind another, there must be something to prop it up.

Challenge your first instincts, too. With the trickiest shapes (like the L blocks), it can seem like there’s more than one way to form the stack. There is not. If your first idea isn’t working, try to look at it with fresh eyes and picture how else a block might be positioned.

Use the edges

Smaller pieces have their edges in black, but they aren’t cut up with black lines. That means, if you think you’ve got a piece oriented correctly but there’s a suspicious black line on one face, the piece is probably doing something you’ve not spotted.

When does this come up?

Cube counting appears in 11+ non-verbal or spatial reasoning papers, including the ISEB Pre-Test. Questions range from simpler stacks of identical cubes to much more complicated constructions, and as ever, the ISEB is adaptive, so if you’re doing well, you may be more likely to get a challenging stack. However, don’t read too much into it. If you don’t get a question type, this could’ve been due to random chance. You will not encounter every type of question in your exam.

How to use this tool

Select a piece from which to build your stack, then place pieces on the stage. You can rotate the pieces or stand them up, but if the piece is greyed out, that means you’re trying to place it in a way that not all of its base is being supported. So, build the stack layer by layer.

There are some tools to help you:

• X-ray will make the cubes transparent, so you can see through the stack.
• Colour by piece will make each block a different colour, so you can better see where one ends and another begins.
• Separate pieces will blow apart the individual pieces so you can visualise how they fit together.
• Focus by layer to see each layer individually. This is best used with cubes, as using it with other pieces will split them apart in a potentially unhelpful manner.

Do not use this tool to answer questions for you before you’ve given them a go yourself. This tool is for making investigations, exploring 3D constructions, and for checking your answers — not for doing it for you!