Dividing a rectangle into maximum number of regions using lines

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At most how many regions can you divide a rectangle in using 6 lines?

I got 16.

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Well, you can get more than 16, And apparently, 22 (image below) is the maximum number. I'd love to see Calvin Lin's proof..

enter image description here

Of course, if the lines don't have to be straight, you can go much higher..