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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?


A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Pythagoras’ Comma

Age 14 to 16
Challenge Level

Here's how Julian from Wilson's School reasoned :

The first note in the scale is 1
The second note is 1 $\times$ 2/3 = 2/3
The third is 2/3 $\times$2/3 = 4/9, but that is less than 1/2, so we multiply it by 2, giving the answer 8/9
The fourth is 8/9 $\times$2/3 = 16/27

This process needs to be repeated 12 times in total.
That means that 1 will be multiplied by 2/3 12 times, so we can do the following calculation:

$(\frac{2}{3})^{12} = 0.0077073466$ (10dp)

We are also know that 7 doublings are required, so 0.0077073466... $\times$ 128 = 0.9865403685 (10dp)

Therefore, they were off by 1 - 0.9865403685... = 0.0134596315 (10dp)

Thanks Julian