Input: Number of divisions
Output: Max number of regions resulted from divisions.
Plan 1: Draw and find patterns. (I don't know how to show that on this slog)
What I noticed immediately is that each line had to interact with all the other lines.
Also, since the divisions were all lines with different slopes, they could only intersect with all the other lines once.
When the new line went from the boundary of the paper to the first division, it split a region in 2 (or "created" a new area).
However, this thinking didn't really go with the empirical findings. I could only find 10 regions when I drew 4 lines. Then I tried it on the computer and it worked.
Divisions - Regions
1 - 2
2 - 4 (2 + 2)
3 - 7 (2 lines + line going through page -1st line, 1st - 2nd, 2nd and page)
4 - 11 (3 lines: page-1st, 1st-2nd, 2nd-3rd, 3rd-page) or 7 + 4
5 - 11 + 5 = 16
I'm going to quickly guess the answer based on this pattern.
If the number of divisions is n, then the number of regions increases by n every time.
Mathematically, this looks something like: 2 + 2 + 3 + 4 + 5 + ... + n
This looks really similar to 1 + 2 + 3 + 4 + ... + n
and is the same as 1 + (1 + 2 + ... + n)
which is 1 + (n* (n+1)) / 2
and that's my answer for now.
Output: Max number of regions resulted from divisions.
Plan 1: Draw and find patterns. (I don't know how to show that on this slog)
What I noticed immediately is that each line had to interact with all the other lines.
Also, since the divisions were all lines with different slopes, they could only intersect with all the other lines once.
When the new line went from the boundary of the paper to the first division, it split a region in 2 (or "created" a new area).
However, this thinking didn't really go with the empirical findings. I could only find 10 regions when I drew 4 lines. Then I tried it on the computer and it worked.
1 - 2
2 - 4 (2 + 2)
3 - 7 (2 lines + line going through page -1st line, 1st - 2nd, 2nd and page)
4 - 11 (3 lines: page-1st, 1st-2nd, 2nd-3rd, 3rd-page) or 7 + 4
5 - 11 + 5 = 16
I'm going to quickly guess the answer based on this pattern.
If the number of divisions is n, then the number of regions increases by n every time.
Mathematically, this looks something like: 2 + 2 + 3 + 4 + 5 + ... + n
This looks really similar to 1 + 2 + 3 + 4 + ... + n
and is the same as 1 + (1 + 2 + ... + n)
which is 1 + (n* (n+1)) / 2
and that's my answer for now.
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