Advent calendar 2023
23 December
There are 18 ways to split a 3 by 3 square into 3 rectangles whose sides all have integer length:
How many ways are there to split a 10 by 10 square into 3 rectangles whose sides all have integer length?
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The square is split into 3 rectangles by drawing two lines on the rectangle. There are two cases: the lines can both go
in the same direction; or the lines go in perpendicular directions, with one going all the way across the square and one going from an edge of the square to the other line.
For an \(n\) by \(n\) square, there are:
- \((n-1)(n-2)/2\) ways to pick two dividing lines that are both vertical;
- \((n-1)(n-2)/2\) ways to pick two dividing lines that are both horizontal;
- \(2(n-1)(n-1)\) ways to pick two dividing lines where one is vertical and goes all the way across the square, and the other is horizontal.
- \(2(n-1)(n-1)\) ways to pick two dividing lines where one is horizontal and goes all the way across the square, and the other is vertical;
In total this makes \((n-1)(5n-6)\) ways to split the square. (10–1)×(5×10-6) is 396.
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