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Weekly Problem 2 - 2013

Weekly Problem 2 -2013

Cubes Within Cubes Revisited

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Imagine you have an unlimited supply of interlocking cubes (all the same size) in different colours. Now imagine starting with one yellow cube. This is covered all over with a single layer of red cubes:
(See animation)
diagram of one yellow cube being covered with a layer of red cubes with a pile of blue cubes to the side, not yet used

This is then covered with a layer of blue cubes.

How many red cubes have you used?

How many blue cubes have you used?

Find an easy way of working this out.

Now imagine adding a layer of green cubes.

How many green cubes are needed?

Have you a quick method for working out the amounts that you would need for each layer?

piles of different coloured cubes

Martha thought about the third layer like this:

  • The whole of the new cube is a $5$ by $5$ by $5$ cube.
  • The inner layers made a $3$ by $3$ by $3$ cube.
  • So the new layer is $5^3 - 3^3 = 125 - 27$ cubes.

Can you see it like this?

Using this method, if you have an $x$ by $x$ by $x$ cube, can you explain how to find an expression for the number of cubes needed to make the next layer?

Emma made the third layer like this:

  • First she added a $5$ by $5$ plinth to the bottom, and another to the top.
  • Then she added a $3$ by $3$ block to each vertical face, and finally four columns of $3$ cubes to the corners.

Can you see it like this?

Using this method, if you have an $x$ by $x$ by $x$ cube, can you explain how to find an expression for the number of cubes needed to make the next layer?

[You may like to view this animation - but at 700kb it is quite large and will take 2 to 3 minutes to download unless you have a broadband connection]

Charlie made the third layer like this:

  • First he added a $3$ by $3$ block to the top and another to the bottom.
  • Then he made lots of pillars of $5$ cubes, and wrapped $16$ of these round the sides

Can you see it like this?

Using this method, if you have an $x$ by $x$ by $x$ cube, can you explain how to find an expression for the number of cubes needed to make the next layer?

[You may like to view this animation - but at 700kb it is quite large and will take 2 to 3 minutes to download unless you have a broadband connection]

Can you show that all three expressions are equivalent?

Can you find another way to think about this problem?

Show how you form your expression, and show that it is equivalent to the examples above.