These grids are filled according to some rules - can you complete them?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

What happens to these capital letters when they are rotated through one half turn, or flipped sideways and from top to bottom?

How would you move the bands on the pegboard to alter these shapes?

These clocks have been reflected in a mirror. What times do they say?

Put a number at the top of the machine and collect a number at the bottom. What do you get? Which numbers get back to themselves?

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

Original text of NRICH problems, as featured in Charlie and Alison's BCME session 2018.

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

Choose some complex numbers and mark them by points on a graph. Multiply your numbers by i once, twice, three times, four times, ..., n times? What happens?

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.