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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.

From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

Six Notes All Nice Ratios

Age 14 to 16 Challenge Level:

Hint 1 : Remember, we need nice simple ratios between any two strings in your scale.

For example 5/9 of the unit would probably sound good with Note 1 and 6/7 would also sound OK with Note 1, but 5/9 and 6/7 would probably not sound good together because their length ratio (35:54) is not simple.

Hint 2 : Once you have the ratios for all your strings - you should be able to show that taking any pair of strings, you only once have a ratio with numbers bigger than 10.

Hint 3: You have ratios based on halves and thirds - what else could you try?