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GCSE coursework cubes problem
By Vix Wyatt on Thursday, October 31, 2002 - 08:13 pm:

im really stuck i have to do my gcse maths coursework. it is to do with 3d sequencing. in involves finding an equation for a 3D cross shape. The equation for the middle layer is n2+(n-1)2. Using the finite series i have to form an equation to solve here is the info i know:
n number of cubes
1 1
2 7
3 25
4 63
5 129

please help me

By Matthew Smith on Thursday, October 31, 2002 - 09:51 pm:

I assume that what you want is a formula for the number of cubes used in the nth cross shape, in terms of n. Here are a few posible lines of attack...

1) You've got a formula for the middle layer. You can probably see that it's going to work for other layers as well (with n suitably changed). You could try adding up these formulae to get one for the whole thing (remembering that you have to count every layer except the middle one twice - once above and once below). Try it first for a particular value of n, then see if you can make it work for other values of n.

2) It's difficult to try to work out the formula directly, but it helps if you know what sort of equation it's going to be. Think about shapes in two dimensions: the square and triangular numbers, and the middle layers of your cross shapes. What sort of formulae do they have? What powers of n are involved when you multiply them out? And how do you think this is going to change for three dimensional shapes, like cubes, or the complete cross shapes?

3) Do you know about the difference method, where you fond the differences between successive terms, and the differences between the differences, and so on? It only works with certain types of sequence, but I think it should work with this one. You might need to work out the number of little cubes in n=6 and n=7 to make sure that the differences are doing what you think they are.

Again, the difference method can tell you about what sort of equation the formula is. You then need to work out a way of getting the individual numbers in the formula. Considering what the 'zeroth' term would be might help get you started.

Remember that, with coursework, you must declare any help you get on the form that you fill in to go with the coursework, and it's a good idea to include a print-out of this text. I've been so cryptic that you shouldn't loose any marks for doing it! If you're still stuck, post again, and I, or someone more helpful, will think up more hints.

Good luck!

By Chris Tynan on Friday, November 01, 2002 - 07:34 pm:

Vix, hint: think about n3 and look at the pattern.

Also, having done GCSE coursework last year, i know providing a good proof of the result and explanation is a good way to gain a couple more marks [as well as a good thing to get used to for A-level and in general]

Good Luck!

Chris