# Math Help - 'Plane' proof

1. ## 'Plane' proof

Consider n planes in the 3-dimensional space in general position (no two are parallel, any three have exactly one point in common, and no four have a common point). What is the number of regions into which these planes partition the space?

Plz do post with detail coz this question's bugging me.
thanks!!

2. Originally Posted by asc
Consider n planes in the 3-dimensional space in general position (no two are parallel, any three have exactly one point in common, and no four have a common point). What is the number of regions into which these planes partition the space?

Plz do post with detail coz this question's bugging me.
thanks!!
Let's see, I will not give a solution here but tell you an idea. Try the idea and tell me if you have problems.
You must be learning this in recurrence relations.
Have you solved it for 2D??
That is...
Given a plane and 'n' lines on it(of which no two are parallel and no three meet at the same point), tell me how many regions does it divide the plane into???

Hint: Imagine the plane being divided by 'n-1' lines and it has $L_{n-1}$ regions. Now ask yourself how many additional regions will the new line divide our old regions into?? Say it is 'k' (hey you have to find this )
then $L_{n} = L_{n-1} + k$. Now solve the relation.
So find 'k' on your own.

If you do this, extending it to 3D involves nearly the same procedure

So let me know your progress

3. Sorry for the double post (to the board)
But it seems you have not taken my suggestion,

Ok now let us do a even simpler problem,

Consider a straight line given to you, say there are 'n' points on it (of which no two are co-incident). Tell me how many line segments does this divide the line into??

It's pretty obvious without recurrence relations, but let us do this with it.
Say n-1 points divide the line into P(n-1) segments, then if I add a new point to this line, we have to consider two cases,
1) In case it is between two existing points, we see(draw a figure and see, please) it divides the region between it into two. Thus one new region added!
2)If it's not between two points, there must still be an existing point(call it Q) close to this new point. Now note that this point divides the region after Q(or before Q) into two regions. So again an additional region.

So we conclude ,in any case 1 extra region for 1 point added.
Thus,
P(n) = P(n-1) + 1; P(0) = 1;
Now solve this
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Questions:
I want to know
If you want to >learn< how to do this?
OR
Just get the proof of the problem?
Tell me and I will either continue helping you or just finish this post with a proof.
If I tell you the proof you will say, "Oh darn! it was so simple" (If you refer "Concrete Mathematics" by Knuth, the proof is in first chapter after Tower of Hanoi problem). But discovering the solution to this problem was a pleasant exercise to me and that is why I was trying to explain it this way.

I believe in
"Give a man a fish and it will feed him for a day,
Teach a man fishing, It will feed him for life"

For maths( and generally)
" Solve a student's problem and he will be satisfied for a day,
Teach a student problem-solving, he will be independently happy forever"

Unfortunately I no longer believe this is true
These days I notice it is more like
" Solve a student's problem and he will be satisfied for a day,
Give him a FORUM, Ah!! in vain, he will come there everyday!!"

Bah! I crib a lot, anyway do tell me if you solve this problem(or if you want a proof)
I am happy to help either way

4. Oh great i got this one. Thanks!!

5. Did you get the proof?? What did you use?