Math solution for number of possible window monitor positions using 1, 2, or 3 divisions

Hello,

What math equation describes the number of possible window positions when dividing a computer screen (rectangle) using one, two, & three sections in one or both (horizontal & vertical) axes?

For example, one division obviously has 1 possibility:
Fig. 1And it appears that with one or two divisions in each direction there are 4 + 2 + 2 = 8 possibilities:
Fig. 2Fig. 3But what about with one, two, or three divisions? I initially counted 9 + 3 +3 + 4 + 12 = 31
Fig. 4Fig. 5Fig. 6Fig. 7but then I realized I had forgotten the 2-3 combinations so that's 12 more (ie 31 + 12 = 43).
Fig. 8I suspect there may be others because 43 seems like a bit of an odd total. That's why it needs math!

And what about having up to four divisions? I didn't even try to start doing that graphically.

Thanks
WeThotUWasAToadAsked:
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sarabandeConnect With a Mentor Commented:
the problem is not well defined yet. why should rectangles (0, 0, 8, 12) and (4, 0, 8, 12) be valid solutions (3rd and 5th yellow) but (0, 0, 6, 12) is not?

if you say all windows must have same size and a solution covers the whole screen, you nicely have

-           0 split lines 1 window
-           1 split line   3 windows
- up to 2 split lines 5 windows
- up to 3 split lines 7 windows

Sara
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Rob HensonConnect With a Mentor Finance AnalystCommented:
Is it not a simple mathematical function:

(Vertical Dividers +1) x (Horizontal Dividers +1)

Examples:
Vert      Horiz
0+1  x   0+1 = 1
1+1  x   0+1 = 2
2+1  x   1+1 = 6

Thanks
Rob H
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WeThotUWasAToadAuthor Commented:
Thanks for the responses.

the problem is not well defined yet. why should rectangles (0, 0, 8, 12) and (4, 0, 8, 12) be valid solutions (3rd and 5th yellow) but (0, 0, 6, 12) is not?
Apologies for the ambiguity in my OP. I did state that the possibilities for a given number of divisions includes those for a lower number but I neglected to account for the lower numbers and create a total . In other words:

1 division = 1 possibility
2 divisions = 1 (from 1 division) + 8 (as stated in OP) = 9 total possibilities
3 divisions = 1 (from 1 division) + 8 (from 2 divisions) + 43* (as stated in OP) = 52 total possibilities  

To avoid the same head scratching among other experts, I think the best step is to award points and close this thread and then re-post the question in a new thread.

Thanks again for the input.
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WeThotUWasAToadAuthor Commented:
See new thread.
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