Tricky maths question: URGENT Can you solve the Mystic Rose? Are you a whizz with algebra?

Question

Help: can you help me come up with a formula for the number of regions?
The "missing" region

Mark a number of points unevenly round a circle and join every point to every other point. How many regions has the circle been divided into? Can you find a formula to predict the number of regions Y given the number of points X?

The diagram alongside shows that there are Y = 8 regions created with X = 4 points. The table below gives some more values which you can check if you like!
Points X 2 3 4 5 6 7
Regions Y 2 4 8 16 31 57
Differences D 2 4 8 15 26

Can you find a (polynomial) function for Y(X) ?

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I need the answer to this

a clue, the number of lines is found here:

The Mystic Rose diagram lets you see how to find the formula N * (N-1) /2 that calucates these numbers
http://bodmas.org/blog/?p=312

Answer 1

It's quite complicated. The link given goes into detail about an inductive method of obtaining this. If you just want the answer it is -

(n^4 - 6n^3 + 23n^2 -18n + 24) / 24

Answer 2

I'm not sure about the first problem because there are only ONE type of ticket given, so you don't know the price of the other ticket(s) (if any) and therefore can't find the amount for each kind of ticket. ----- For the second problem, it can be confusing if you don't know how: The way I would do it is to half 72 first because the problem states "the sum of TWO numbers". Halving 72 gives you 36. Since one of the two numbers is 8 more than the other, I added half of 8 (4) to one of the 36's (since there are two 36's in 72), and subtracted half of 8 (4 again) from the other 36: 72 / 2 = 36 Splitting 72 into two equal amounts 36 + 36 = 72 Showing that the sum of the two numbers is 72 +4 -4 Adding 8(0.5) to one 36, subtracting 8(0.5) from the other. 40 + 32 = 72 The numbers are 40 and 32 because 40 is 8 more than 32, and 40 + 32 = 72. ~Jeff