1. Two views
Here are two views of the same shape made up of coloured cubes. How many cubes are there in the shape? What’s the minimum number? Maximum number? What does the side view look like?

2. Three views
Here’s another bit of information – view from the left. Have you now got enough information to know what the shape looks like?

3. Four views
A fourth view gives us all the information we will need to make the shape -

4. The Solution
Here is a picture of the shape itself.

5. What are we looking at? Front view Top view
Here are the front view and top view of something – what might the side view look like?
Top view

6. A side view
We see the full picture.

7. Pentominoes
Another 2-D example. Here are the 12 pentominoes – arrangements of 5 squares. Can you find two different pentominoes which will fit together to cover a 2 by 5 grid? Why can’t you find a pair that work? (You would need repeats of the same shape like the L-shape on the top row.) See next slide.

8. Open cubes
Another 2-D example. Here are the 12 pentominoes – arrangements of 5 squares. Can you find two different pentominoes which will fit together to cover a 2 by 5 grid? Why can’t you find a pair that work? (You would need repeats of the same shape like the L-shape on the top row.) See next slide.

9. Pentominoes
Another 2-D example. Here are the 12 pentominoes – arrangements of 5 squares. Can you find two different pentominoes which will fit together to cover a 2 by 5 grid? Why can’t you find a pair that work? (You would need repeats of the same shape like the L-shape on the top row.) See next slide.

10. 2 x 5 rectangles
Some of the pentominoes will fill the 2X5 grid with a copy of themselves, others will either not fit into the rectangle or will split the remaining squares into two groups as in the third example here.
Now try to find three different pentominoes which will fill a 3X5 grid.

11. 3 x 5 rectangle
It is not possible since one of the blocks must take the 1X5 shape leaving a 2X5 rectangle still to fill. We have just found that we cannot fill this with two different shapes. Hence we have proved that we can not put all the shapes into 4 different 3X5 rectangles.

12. Imagine this cube
What colours will border blue? Will border red? Will share an edge with green? If pink is on the top what is on the bottom? What colour is opposite brown? There are many ways to make a net for a cube.

13. Same cube - different net
Here is another possible net of the cube. Colour it so that when the two nets are folded, the cubes look identical.

14. One solution
Here is another possible net of the cube. Colour it so that when the two nets are folded, the cubes look identical.

15. Nets of cubes? A B D C
Which of these shapes made up of six squares (hexominoes) can be folded to make cubes? (A, C AND D) F E

16. Which of these dice is correct?
Which of the dice shown is produced when you fold up the net?

17. The pop-up cube
Using a sheet of A4 paper or thin card, try to construct this pop-up cube to work like a pop-up birthday card.

18. but does not pop out when closed!
Make sure it pops up
but does not pop out when closed!
Using a sheet of A4 paper or thin card, try to construct this pop-up cube to work like a pop-up birthday card.

19. A further challenge

20. Another net
What special name is given to the type of triangles? What solid shape can be made by folding this net? How many faces will it have? How many corners? How many edges?

21. Regular tetrahedron

22. Views from above
Here are pictures looking down on a tetrahedron. The shape has been rolled around so that each of its corners has been shown at the top. One of the pictures cannot be right – which is it? How do you know? What should it look like?(C is incorrect)

23. View from the front ?

24. View from the left ?

25. View from the right ?

26. View from above ?