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Building 3D shapes

Pieces made of cubes build a model, and you pick the set that fits. Here’s how to count, fit and check every time, even when the model is turned round.

Rush doesn’t practise building 3D shapes yet. These games practise related skills: Top View and Missing Piece.

Start the guide Practise 8 questions

Chapter 1

What are building 3D shapes questions?

You see a model built from cubes, and some sets of pieces. Each piece is a few cubes stuck together face to face, like toy bricks. You pick the set that builds the model exactly.

  1. The model. Cubes stacked on a flat base, drawn at an angle. Some hide behind others.

  2. The sets. Four here, numbered 1 to 4. Exam papers usually give five, A to E.

  3. The rule. Use every piece once, with no cubes left over and no gaps. Turn a piece any way you like, but it can’t bend or come apart.

Some questions swap it round: they give you the pieces and ask which model they build. The same steps work.

Which set of pieces builds this model? Each piece is used once and can be turned any way.

The sets of pieces

  1. 1
  2. 2
  3. 3
  4. 4

The seven pieces you’ll meet

Know each one’s name and how many cubes it has. The count is your first check, every time.

2 cubes

  • Domino

3 cubes

  • Straight three
  • L of three

4 cubes

  • Square
  • T of four
  • L of four
  • Zigzag of four

Chapter 2

Solve it in 4 steps.

Here’s the question. More than one set can have the right number of cubes, so check every set.

The question

Which set of pieces builds this model? Each piece is used once and can be turned any way.

The sets of pieces

  1. 1
  2. 2
  3. 3
  4. 4

  1. Step 1

    Count the model, hidden cubes too.

    Count it stack by stack, as on the counting cubes page: back row 1, 2, 2 and front row 1, 1. That’s 7. Only 6 show. The bottom cube of the back-middle stack hides behind the cube in front of it.

    6 show, 7 in all.

  2. Step 2

    Add up each set.

    Count the cubes in each piece and add. Set 1 is two Ls of three: 3 + 3 = 6, so it’s out. Sets 2, 3 and 4 all make 7, so the count can’t choose between them.

    • Set 1 3 + 3 = 6
    • Set 2 3 + 4 = 7
    • Set 3 3 + 4 = 7
    • Set 4 3 + 4 = 7

    Only set 1 has the wrong total.

  3. Step 3

    Fit the most awkward piece first.

    The longest or biggest piece has the fewest places to go. Set 2’s straight three needs three cubes in a line, and this model has only one: along the bottom of the back row, through the hidden cube.

    The only line of three.

  4. Step 4

    Check that what’s left is the other piece.

    The cubes left must all touch and make exactly the other piece. Set 2 leaves two pairs that don’t touch: out. Set 4’s T needs the same line and leaves two parts too: out. In set 3 the square stands up at the back right, and the three cubes left make an L.

    Set 2: two pairs left, no square
    Set 3: the square and the L fit

Three mistakes it sets

The model from the question.
The mistakeDo this instead
Counting only the cubes you see. Set 1’s two Ls make 6, exactly the cubes that show. It’s there for anyone who misses the hidden cube.Count every stack down to the floor. The model has 7.
Stopping when the total matches. Set 2 makes 7, but its straight three leaves two pairs that don’t touch.Fit the pieces in before you choose.
Counting what’s left, not looking at it. Set 4’s T fits and leaves 3 cubes, the size of the L. But they’re in two parts.What’s left must be one part, in the other piece’s shape.

So it’s set 3. It has 7 cubes, and the square and the L fill the model with nothing left over.

Remember: Count. Hardest first. Check what’s left.

Chapter 3

When it gets harder.

Pieces drawn a different way round, cubes that stick out, and three pieces at once. Four ideas crack them.

  • Turn it, but never bend it

    Lying flat, standing up or upside down, it’s the same piece.

    These are all one L of three. A straight three can stand up too, but it never becomes an L. Trap: crossing out a set because its L lies flat and the one in the model stands up.

  • A cube that sticks out

    A cube that touches only one other cube must be in the same piece as it.

    The top cube here touches only the cube under it, so those two are one piece: the domino. The four left make the square. Trap: putting the domino anywhere else, which leaves the top cube with nowhere to go.

  • Try every place

    If what’s left splits in two, try the piece somewhere else before crossing the set out.

    Lying along the front, this straight three leaves a lone cube and the top of the tower. Standing up as the tower, it leaves an L, so a straight three and an L build it. Trap: giving up after the first try.

  • Three pieces, one hidden cube

    Add all three, fit the awkward ones first, and the last piece is what’s left.

    3 + 3 + 2 = 8, and the model has 8, one hiding at the back. The straight three fits only along the front. The L takes the back, hidden cube and all, and the domino lies on top. Trap: forgetting that a hidden cube is part of a piece too.

Chapter 4: In the exam

Turned models and missing pieces.

“Which set builds this model?” works in the exam just as in chapter 2. Two more forms need a little extra.

Which model is this one, turned round?

  1. A
  2. B
  3. C
  4. D (the answer)
  5. E

  1. Check 1

    Count the cubes.

    Turning a model never changes how many cubes it has. The model has 5. B has 6: out.

  2. Check 2

    Find the same parts.

    The model is an arm of three with a tower at one end and a cube sticking out at the other. In C the cube sticks out from the middle: out.

  1. Check 3

    Which side does it stick out?

    Stand at the tower and walk along the arm. In the model, the end cube is on your left. In A and E it’s on your right: they’re mirror images. No turn can make the mirror image of a lopsided model, so they’re out.

    The model: on your left
    E: on your right

So it’s D. It has 5 cubes, a tower at one end of the arm, and the end cube on your left as you walk from the tower.

Remember: Same count, same parts, same side.

Which piece fills the gap?

Which piece completes a block 3 cubes wide, 2 deep and 2 tall?

  1. A
  2. B
  3. C (the answer)
  4. D
  5. E
The gap: an L of four.
  1. Count. A full block has 3 × 2 × 2 = 12 cubes. The model has 6 on the bottom and 2 on top, so 8. The piece needs 12 − 8 = 4 cubes: D, with 3, is out.
  2. Look at the gap. It’s the whole top of the front row, and one more cube behind its right-hand end: an L of four.
  3. Find that piece, turned. C is an L of four standing up. Lie it down and it fills the gap. A, B and E have 4 cubes, but the wrong shape. So it’s C.

Other ways the exam asks it

  • “Which model can these pieces build?” Chapter 2 the other way round: add up the pieces, cross out the models with a different count, then fit the hardest piece in each one left.
  • “Which set can’t build it?” or “Which piece isn’t needed?” Underline the “not” first. The same checks find the one that fails.
  • “The pieces have not been turned.” Some papers say this. Then each piece must sit in the model exactly as it’s drawn, which rules sets out faster.
  • Shaded or coloured pieces. Each piece’s colour must match its part of the model too.

Chapter 5

Try it yourself.

Easier ones first. Try each one before you open its answer.

  1. Question 1Level 1

    Which set of pieces builds this model? Each piece is used once and can be turned any way.

    The sets of pieces

    1. 1
    2. 2
    3. 3
    4. 4
    Show answer

    Answer: Set 3.

    How to answer it

    1. Count the model: the back stacks are 2 and 2, and the front-left cube 1. That is 5 cubes, though only 4 show: the bottom cube at the back left is hidden behind the front cube.
    2. Set 2 has 6 cubes and set 4 has 4. Both are out.
    3. Set 1 has 5, but the model is only 2 cubes each way, so the straight piece of three can't fit.
    4. Set 3: the L stands on the left, up the back-left stack and along to the front cube, and the domino stands up at the back right. The answer is set 3.
    Why the other answers are wrong
    • Set 1The model is only 2 cubes wide, 2 deep and 2 tall, so there is no room for a straight piece of three.
    • Set 2Two Ls make 6 cubes. The model has 5.
    • Set 4Two dominoes make 4 cubes, the number you can see. The model has 5, with one hidden at the back left.

  2. Question 2Level 1

    Which model can be built from these pieces? Each piece is used once and can be turned any way.

    The pieces

    The models

    1. 1
    2. 2
    3. 3
    4. 4
    Show answer

    Answer: Model 2.

    How to answer it

    1. The pieces have 3 + 3 = 6 cubes. Model 4 has 5, so it is out.
    2. Model 3 has 6 cubes, one of them hidden, but it is only 2 cubes in every direction, so the straight piece can't fit.
    3. Model 1 is two rows of three. The straight piece fits along either row, but the cubes left are the other row, a second straight piece, not an L.
    4. Model 2: the straight piece runs along the bottom of the back row, and the L stands at the right, made of the front-right stack and the cube on top of the back-right stack. The answer is model 2.
    Why the other answers are wrong
    • Model 1Wherever the straight piece goes, the three cubes left are a straight row too, not an L.
    • Model 3It is only 2 cubes in every direction, so there is nowhere for the straight piece of three.
    • Model 4It has 5 cubes. The pieces have 6.

  3. Question 3Level 1

    Which model can be built from these pieces? Each piece is used once and can be turned any way.

    The pieces

    The models

    1. 1
    2. 2
    3. 3
    4. 4
    Show answer

    Answer: Model 2.

    How to answer it

    1. The two Ls have 6 cubes. Model 1 has 5 and model 4 has 7, so both are out.
    2. Model 3 has 6 cubes. Try an L anywhere in it: the three cubes left always fall into two separate parts, so they can't make the second L.
    3. Model 2: one L lies flat across the back-left, back-middle and front-left cubes, and the other stands up at the back right, made of the back-right stack and the cube on top of the back-middle stack. The answer is model 2.
    Why the other answers are wrong
    • Model 1It has 5 cubes. The two Ls have 6.
    • Model 3It has 6 cubes, but wherever the first L goes, the cubes left are in two parts that don't touch.
    • Model 4It has 7 cubes. The two Ls have 6.

  4. Question 4Level 2

    Which model can be built from these pieces? Each piece is used once and can be turned any way.

    The pieces

    The models

    1. 1
    2. 2
    3. 3
    4. 4
    Show answer

    Answer: Model 4.

    How to answer it

    1. The pieces have 3 + 4 = 7 cubes. Count each model with its hidden cubes.
    2. Model 2 shows 7 cubes but has 8: the back-left stack is as tall as the front-left one, so its bottom cube is hidden. Model 3 has 6. Both are out.
    3. Model 1 has 7, with one hidden under the back-middle top. The square fits only flat on the floor, on the left half or the right half, and each time the three cubes left are in two parts.
    4. Model 4: the square stands up across the back-left and back-middle stacks, and the L stands at the right, made of the back-right stack and the front-right cube. The answer is model 4.
    Why the other answers are wrong
    • Model 1It has 7 cubes, but the square fits only flat on the floor, at the left or the right, and either way the cubes left are in two parts.
    • Model 2It shows 7 cubes, but the back-left stack matches the height of the one in front, which hides an eighth cube.
    • Model 3It has 6 cubes, one of them hidden. The pieces have 7.

  5. Question 5Level 2

    Which set of pieces builds this model? Each piece is used once and can be turned any way.

    The sets of pieces

    1. 1
    2. 2
    3. 3
    4. 4
    Show answer

    Answer: Set 2.

    How to answer it

    1. Count the model: back row 2, 2, 1 and front row 1, 1, 1. That is 8 cubes, though only 6 show: the back-left and back-middle stacks each hide a cube underneath.
    2. Set 3 has 3 + 4 = 7 cubes, so it is out.
    3. Set 1: a square fits in three places, standing up at the back or lying flat at the left or right end of the floor. Each time, the four cubes left aren't a square.
    4. Set 4: a T fits in three places, and each time the cubes left fall into separate parts.
    5. Set 2: the square stands up across the back-left and back-middle stacks, and the L of four lies flat along the front row and round to the back-right cube. The answer is set 2.
    Why the other answers are wrong
    • Set 1A square fits standing up at the back or flat at either end of the floor, and each time the four cubes left aren't a second square.
    • Set 3An L of three and a square make 7 cubes. The model has 8, two of them hidden.
    • Set 4The T fits three ways, and each way splits the cubes left into separate parts.

  6. Question 6Level 2

    Which model can be built from these pieces? Each piece is used once and can be turned any way.

    The pieces

    The models

    1. 1
    2. 2
    3. 3
    4. 4
    Show answer

    Answer: Model 3.

    How to answer it

    1. The pieces have 4 + 2 = 6 cubes.
    2. Model 4 shows 6 cubes but has 7: a middle-left stack of 2 stands in front of the back-left stack of 2 and hides its lower cube. Out.
    3. Model 1 is only 2 cubes in every direction, and the zigzag is three cubes long, so it can't fit.
    4. Model 2 is a flat block of six. The zigzag fits two ways, and each time it leaves two corner cubes that don't touch.
    5. Model 3: the zigzag lies flat across the back-left, back-middle, front-middle and front-right cubes, and the domino stands up at the back right. The answer is model 3.
    Why the other answers are wrong
    • Model 1The zigzag is three cubes long, and this model is only 2 cubes in every direction.
    • Model 2The zigzag fits two ways, and each leaves two corner cubes on their own, which can't make a domino.
    • Model 4It shows 6 cubes, but the back-left stack hides its bottom cube behind the stack in front: 7 cubes.

  7. Question 7Level 3

    Which set of pieces builds this model? Each piece is used once and can be turned any way.

    The sets of pieces

    1. 1
    2. 2
    3. 3
    4. 4
    Show answer

    Answer: Set 3.

    How to answer it

    1. Count the model: a flat base 3 by 2 with two more cubes on top in the middle. That is 8 cubes, though only 7 show: the bottom cube of the back-middle stack is hidden.
    2. Set 2 has 3 + 4 = 7 cubes, the number you can see, so it is out.
    3. Set 1: a square fits three ways, standing up in the middle or lying flat on the left or right of the base. Each time, the four cubes left are two separate pairs.
    4. Set 4: a zigzag fits two ways, both flat on the base, and each leaves three separate parts.
    5. Set 3: one T stands up along the back row, with the back-middle top as its fourth cube, and the other does the same along the front. The answer is set 3.
    Why the other answers are wrong
    • Set 1A square fits three ways, and each leaves two separate pairs of cubes, not a second square.
    • Set 2An L of three and a square make 7 cubes, the number you can see. The model has 8, with one hidden under the back-middle top.
    • Set 4A zigzag fits two ways, and each leaves the other cubes in three separate parts.

  8. Question 8Level 3

    Which model can be built from these pieces? Each piece is used once and can be turned any way.

    The pieces

    The models

    1. 1
    2. 2
    3. 3
    4. 4
    Show answer

    Answer: Model 2.

    How to answer it

    1. The pieces have 3 + 3 + 2 = 8 cubes.
    2. Model 3 shows 8 cubes but has 9: the back-left stack hides its bottom cube. Model 4 has 7. Both are out.
    3. Model 1 has 8 cubes, but it is a block 2 cubes each way, with no room for the straight piece of three.
    4. Model 2 shows 7 cubes and has 8, with one hidden at the back left. Stand the straight piece up as the tower at the back right, put the L upright in the back-left corner, and lay the domino across the two front cubes. The answer is model 2.
    Why the other answers are wrong
    • Model 1It has 8 cubes, but at 2 by 2 by 2 there is nowhere for a straight piece of three.
    • Model 3It shows 8 cubes, but the back-left stack has a ninth hidden at its foot.
    • Model 4It has 7 cubes, one fewer than the pieces. Only 5 of them show.
Next in non-verbal reasoning Odd one out How to do non-verbal reasoning odd one out questions by finding what four figures share, one feature at a time, and how to answer 'most like' questions. Read the guide

For parents.

What parents ask us about building 3D shapes.

Questions parents ask

Can the pieces be turned over?

Usually, yes. A piece can be turned any way you could turn a real one in your hand, including onto its side or upside down. The only thing it can't do is change shape. Read the question first, though: some say the pieces have not been turned, and then each piece must sit in the model exactly as it is drawn, which rules options out faster.

Does every piece have to be used?

In these questions, yes: the pieces together must make the whole model, with no cubes left over and no gaps. That is why counting the cubes rules out so many options straight away.

My child can't picture the pieces turning. What helps?

Make the pieces. Glue or stick together a few sugar cubes or toy bricks in the shapes of a domino, a straight three, an L and a square, then build the models on this page with them. After a few goes, children start to see where each piece can go without touching anything.

Practise the related skills in Rush.

  • Top ViewIts block models are drawn like the ones here, so it gives practice in reading how tall each stack is and which row it stands in.
  • Missing PieceChoosing the piece that fills a gap exactly, without flipping it, is the flat version of fitting a piece into a model.

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