Question

all the six faces of a cube are extended in all directions the number of regions into which the whole space is divided by these six planes

Answer 1

In the easiest case, 1 dimension, suppose you have a line and you ask

how many regions it can be divided into by n points. That's easy: each

point divides one of the existing segments into 2, so each new point

adds 1, so the total is n+1. (When n is zero, there's one line; adding

a point makes 2 half-lines, and so on.)

Now look at 2 dimensions - a plane divided up by n lines. By messing

around, you can see that for n = 0, 1, 2, 3, the answers are: 1, 2, 4,

7 regions, right? Well, suppose you've worked it out for some number

n of lines, and you add the next line. In general, it will hit all n

lines, so if you look at the intersections of the old lines with your

new one, it gets hit n times, right? So the new line is divided into

n+1 segments. (We just worked this out in the previous paragraph).

So each of those n+1 segments will divide an existing region into two

regions, so there will be n+1 new regions created, so you can work out

the number of 2-D regions you get with n lines:

1 + 1 + 2 + ... + (n+1)

(The initial "1" is because there's already one region when you start.)

So the values from the formula above are:

lines regions

0 1 = 1

1 1+1 = 2

2 1+1+2 = 4

3 1+1+2+3 = 7

4 1+1+2+3+4 = 11

5 1+1+2+3+4+5 = 16

...

Now go to three dimensions. Assume you know the answer for n planes,

and you want the answer for n+1. Well, the (n+1)st plane will hit all

the n planes in one line each, so that plane is hacked into the number

of regions that n lines will create, which we just worked out as the

sum above.

Each of those plane regions will divide a volumetric region into 2

pieces, so the answers for 3 dimensions are:

planes regions

0 1 = 1

1 1+1 = 2

2 1+1+2 = 4

3 1+1+2+4 = 8

4 1+1+2+4+7 = 15

5 1+1+2+4+7+11 = 26

6 1+1+2+4+7+11+16 = 42

...

Let me list all the results above:

number of points, lines, planes:

0 1 2 3 4 5 6 7

dim --------------------------

1 1 2 3 4 5 6 7 8 ...

2 1 2 4 7 11 16 22 29 ...

3 1 2 4 8 15 26 42 64 ...

The first row is obvious - to get any other number in the chart, add

together the number to it's left to the number above it.

You can actually find a formula for it if you like. For dimension 1,

it's linear: n+1. For dimension 2, it'll be quadratic, and for

dimension 3, cubic. You can find a good reference for finding the

formula for dimension 2 at:

Answer 2

Step-by-step explanation:

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