3D shapes: counting cubes: 11 Plus non-verbal reasoning practice
How many cubes make a 3D shape, including the hidden ones.
Less commonIn some papers only, mainly spatial sections and a few independent schoolsAbout 45 seconds a question in the test
The big idea
Every cube must rest on the ground or on another cube. So if you can see a cube at the top of a column, there must be cubes underneath it, even when you cannot see them.
Count one column at a time: find the height of each column from the top, then add the heights together.
Tips that save marks
Work from the back row to the front.
A cube you cannot see still counts if it is holding one up.
Check your total by counting the cubes you can see, then adding the hidden ones.
A worked example
Three columns of height 3, 2 and 1 stand in a row. How many cubes?
Add the column heights.
3 + 2 + 1 = 6.
Six cubes, even though the bottom ones are hidden.
Now try five
Try each one before you look at the answer: answering from memory is what makes it stick. Tap an option to check it. Most children get three or four right at first; five out of five means move on to the next topic.
Question 1Aim for about 45 seconds
How many cubes make this shape? Every cube is resting on the ground or on another cube.
Show the worked answer
Answer: A, 5
How to work it out
Count each column from the back: 2 + 1, then 1 + 1.
That makes 5.
Only 4 can be seen; the rest are hidden underneath.
Why the other options are wrong
B (7): Too many: no cube is hidden unless it holds another up.
C (6): One too many. Count column by column, from the back.
D (4): This counts only the cubes you can see. Some cubes are hidden underneath, holding the others up.
E (3): Too few: count every cube in each column, down to the ground.
Question 2Aim for about 45 seconds
How many cubes make this shape? Every cube is resting on the ground or on another cube.
Show the worked answer
Answer: C, 11
How to work it out
Count each column from the back: 3 + 3 + 2, then 1 + 1 + 1.
That makes 11.
Only 9 can be seen; the rest are hidden underneath.
Why the other options are wrong
A (10): One too few. Did you miss a hidden cube at the bottom of a column?
B (6): This counts the columns (the tops you can see from above), not the cubes in each column.
D (9): This counts only the cubes you can see. Some cubes are hidden underneath, holding the others up.
E (12): One too many. Count column by column, from the back.
Question 3Aim for about 45 seconds
How many cubes make this shape? Every cube is resting on the ground or on another cube.
Show the worked answer
Answer: C, 6
How to work it out
Count each column from the back: 2 + 1, then 2, then 1.
That makes 6.
Only 5 can be seen; the rest are hidden underneath.
Why the other options are wrong
A (4): This counts the columns (the tops you can see from above), not the cubes in each column.
B (8): Too many: no cube is hidden unless it holds another up.
D (7): One too many. Count column by column, from the back.
E (5): This counts only the cubes you can see. Some cubes are hidden underneath, holding the others up.
Question 4Aim for about 45 seconds
How many cubes make this shape? Every cube is resting on the ground or on another cube.
Show the worked answer
Answer: E, 12
How to work it out
Count each column from the back: 2 + 2 + 2, then 2 + 1, then 2 + 1.
That makes 12.
Only 9 can be seen; the rest are hidden underneath.
Why the other options are wrong
A (7): This counts the columns (the tops you can see from above), not the cubes in each column.
B (9): This counts only the cubes you can see. Some cubes are hidden underneath, holding the others up.
C (11): One too few. Did you miss a hidden cube at the bottom of a column?
D (13): One too many. Count column by column, from the back.
Question 5Aim for about 45 seconds
How many cubes make this shape? Every cube is resting on the ground or on another cube.
Show the worked answer
Answer: C, 8
How to work it out
Count each column from the back: 2 + 1 + 1, then 2, then 2.
That makes 8.
Only 7 can be seen; the rest are hidden underneath.
Why the other options are wrong
A (7): This counts only the cubes you can see. Some cubes are hidden underneath, holding the others up.
B (5): This counts the columns (the tops you can see from above), not the cubes in each column.
D (9): One too many. Count column by column, from the back.
E (10): Too many: no cube is hidden unless it holds another up.
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