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A Problem of Time

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

Quad in Quad

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Hand Swap

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the minute hand and hour hand had swopped places. What time did the train leave London and how long did the journey take?


Age 14 to 16
Challenge Level

ABCD is a non-convex quadrilateral. The points P, Q, R and S are the midpoints of the edges of ABCD. You can change the shape of the quadrilateral ABCD.

What do you notice about the quadrilateral PQRS and its area as ABCD changes?

If you say that what you have noticed is always true, then you are making a conjecture. Can you prove your conjecture?