Symmetry

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?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

I noticed this about streamers that have rotation symmetry : if there was one centre of rotation there always seems to be a second centre that also worked. Can you find a design that has only one centre of rotation ? Or if you thought that was impossible, could you say why ?

What groups of transformations map a regular pentagon to itself?

Can you devise a fair scoring system when dice land edge-up or corner-up?

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

Find the shape and symmetries of the two pieces of this cut cube.