Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Intersecting Squares

Stage: 3 and 4 Short Challenge Level:

Each of the overlapping areas contributes to the area of exactly two squares. So the total area of the three squares is equal to the area of the non-overlapping parts of the squares plus twice the total of the three overlapping areas, i.e. $(117 + 2(2 + 5 + 8))\;\mathrm{cm}^2 = (117 + 30)\;\mathrm{cm}^2 = 147\;\mathrm{cm}^2$.

So the area of each square is $(147 \div 3)\;\mathrm{cm}^2 = 49\;\mathrm{cm}^2$. Therefore the length of the side of each square is $7\;\mathrm{cm}$.

This problem is taken from the UKMT Mathematical Challenges.
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