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If we consider one quadrant of the square divided into a grid of 6×6 squares, we see that the area of the octagon is 6/36 = 1/6 the area of the square. We also see that each side of the octagon is √5 times the side of one of the small squares in the grid.

If the large square has area

Athen the perimeter of the octagon isp = (2/3)(√5)(√A).So:

p²= 20A/9It follows that

Amust be a multiple of 9. So there are only 11 candidates forA.If

A= 9n(n=1 to 11), thenp²= 20n, so 20nmust be a perfect square.We can examine the possibilities by hand (I think it’s pretty obvious what

nmust be for 20nto be a square number), or this Python program runs in 34ms.Solution:The area of the courtyard garden is 45m². The perimeter of the octagonal pool is 10m.