How many rectangles there are in a rectangle that is split horzintally by 12 lines and vertically by 18 lines?

Question

Can you generalize to a J by k lines ?

We are not looking here for simple single reactangles but rather the different recatngles formed by this big mesh.

Answer 1

I can't pull off the j,k expression here. However it would be:
the sum of 0 to j+1 times the sum of 0 to k+1. There is a symbol that does a sumation like that but I can't remember it.
For 13 it comes out to 91, for 19 it comes out to 190.
91*190=17,290

Answer 2

The above answers are wrong because you are only counting the smallest rectangles. However, the lines also create larger rectangles. The largest rectangle is itself a rectangle in the rectangle.

So just to be clear, assuming that the border lines count as 2 of the 12 horizontal lines and 2 of the 18 vertical lines respectively, what are left are 16 vertical and 10 horizontal lines which give 187 (17*11) small rectangles. The largest rectangle is now counted which makes the grand total so far at least 188.

Now if you assume the axiom that all squares are rectangles, then the dimensions of the rectangle are meaningless. Now it's up to some smart guy to figure out how to get the rest of the rectangles.

Answer 3

I believe in general it would be something like...

Sigma x (from 1 to x+1) [Sigma y (from 1 to j+l) (x*y)]

where "Sigma" means add up all the possibilities for x and y from 1 to the number of lines going that way +1. For example, use a rectangle with 2 lines going one way and 1 line going the other way, there should be (1*1)+(1*2)+(2*1)+(2*2)+(3*1)+(3*2) = 1+2+2+4+3+6 = 18 rectangles. I'm not figuring it out for 12 by 18 though! That would be a few hundred terms to add up!

Answer 4

There are
1*1 rectangles of size 13*19
1*2 rectangles of size 13*18
2*1 rectangles of size 12*19
etc.

and in general
a*b rectangles of size (N-a) * (M-b)
where N = J+1 and M = K+1

The total number of rectangles is equal to the sum of a*b over a = 1, .., J+1 and b = 1, ..., K+1:

SUM_a SUM_b a*b
=
(SUM_a a) * (SUM_b b)
=
(J+1)(J+2)/2 * (K+1)(K+2)/2
=
(J+1)(J+2)(K+1)(K+2)/4

In your example, 12*13*19*20/4 = 3705.

Answer 5

There are (j+1)(k+1) such rectangles, in general.
Here, specifically, we have (12+1)(18+1) = 13*19 = 247 rectangles.

Answer 6

Wrong, all wrong.

Dont forget that you can put two, or three or four or any number of your rectangles together to make additional ones. If you could ever get an answer, it would be in the thousands

Answer 7

multiply 13 and 19 to get the answer. More generally:
r = (j + 1)(k + 1)
r = number of rectagles
j = number of horizontal lines
k = number of vertical lines

Answer 8

13x19=247