Following some relatively popular Twitter posting of maths problems, I've decided to start posting a weekly collection of interesting puzzles I have encountered. I'll be posting solutions on the following Monday.

Here's this week's collection, including puzzles from this month's MathsJam:

#### Burning Ropes

You have two ropes and some matches. Each rope, if lit at its end, will burn for 60 minutes. But the rate of burning is not regular, so cutting a rope in half doesn’t result in a burn time of 30 minutes.
How can you use the ropes to time exactly 45 minutes?

#### Pole Position

Two poles stand vertically on level ground. One is 10 feet tall, the other 15 feet tall. If a line is drawn from the top of each pole to the bottom of the other, the two lines intersect at a point 6 feet above the ground. What’s the distance between the poles?

#### Circles

There is a quarter circle with radius 2r and centre A and two semi circles with radius r and centres B and C.

Prove that the red area is equal to the blue area.

#### Pyramid and Tetrahedron

If four equal equilateral triangles form the sides of a square-based pyramid, what is the ratio of the volume of the pyramid to the volume of the tetrahedron whose sides are the four triangles?

#### 8! minutes

How many weeks are there in 8! (8×7×6×5×4×3×2×1) minutes?

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