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It's often time to be giving presents to friends. We usually wrap them up so that there is some fun in unwrapping them. Little baby brothers and sisters tend to like the unwrapping more than the present itself!
We should probably try to save paper, though, and use no more than is necessary. So let's do some wrapping up and never have one piece of paper overlapping another. Remembering this, let's have a look at a very practical challenge.
It would probably be good to use some of those cubes which fit together [multilink] that are around in most schools and put perhaps 27 of these together to make a cube [which in this case would be 3 by 3 by 3].
This is just so that we can have a reasonable sized prezzie, maybe it's a box with an interesting game in it ... I don't know!
We are just concerned with its shape:-
We need to wrap it up, remembering that we can have no overlaps. One rule here is that the paper will be made up of squares the size of the faces of the cube.
So you may decide on something like this:-
You notice that I've used red paper that is red on both sides so that nothing changes if I turn it over.
You could of course find other shapes to use. Here are just two more:-
Now you find the rest!
Remember to avoid reflections or turning over!
Now for part two of this challenge. What happens if we make one of the sides of this cube a bit different so that it becomes a cuboid?
How many different shapes of paper can be used this time?
Try to find them all.
Now for the third part. What if the prezzie were different from a cube in two directions?
Can you predict how many different shapes of wrapping paper there will be this time?
If you can't, then make them all - that's one sure way of finding out!
Well, good luck with this rather different challenge which only becomes an investigation when you now ask the question "I wonder what would happen if ...?''
You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.
In this problem you have to place four by four magic squares on the faces of a cube so that along each edge of the cube the numbers match.
Which of the following cubes can be made from these nets?