Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
How far have these students walked by the time the teacher's car
reaches them after their bus broke down?
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
An ant is crawling in a straight line from one corner of a table to the opposite corner when he bumps into a one centimetre cube a sugar. Instead of crawling round it, or eating his way through it, he climbs straight up and over it before continuing on his intended route. How much does the detour add to the expected length of his journey?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
This problem is taken from the UKMT Mathematical Challenges.