Coloured Glass Solution

25 Sections

It becomes much clearer if you use squares rather than rectangles (the principle remains unchanged) and rotate them regularly, i.e. for 2 rotate one by 45 degrees, for 3 rotate one by 30 and the second by 60. It has been graphically confirmed that 5, 6 and 7 squares give 81, 121 and 169 sections, respectively (i.e. 9^9, 11^11 and 13^13).

As a square has 4 corners, for each square that is laid on top (not aligned with any other square beneath), a further 4 corners will be created. Add one square - add 4 corners, add another square - add another four corners and so on. This creates a pattern of one central section and increasing numbers of sections around it. The interesting point is that the number of sections seems to be equal to the number of squares added to the original square multiplied by the number of corners created plus the central section, i.e.
Start with original square:

1 no squares added, 4 corners, central section = (0*4)+1 = 1 = 1^2
1 square added, 8 corners, central section       = (1*8)+1 = 9 = 3^2
2 squares added, 12 corners, central section    = (2*12)+1 = 25 = 5^2
3 squares added, 16 corners, central section    = (3*16)+1 = 49 = 7^2
4 squares added, 20 corners, central section    = (4*20)+1 = 81 = 9^2
5 squares added, 24 corners, central section    = (5*24)+1 = 121 = 11^2
6 squares added, 28 corners, central section    = (6*28)+1 = 169 = 13^2

This series follows the formula

4n^2 - 4n + 1

Where n is the number of squares.
This generalises very nicely into: xn^2-xn+1

which gives the number of sections produced by n instances of an x-sided regular polygon - certainly for 10>x>2 and 20>n>0, where x and n are positive integers.

Solution by Robin G. Sharman

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