I have two identical 4 x 4 squares as shown, identical except for orientation and position.

Clearly it is possible to move the tilted square to the position of the upright square by first moving it so that one of its corners coincides with that of the upright square, and then to perform a rotation so that the other three corners also coincide. You might also choose to perform a rotation first, and then a move to get the squares to line up.

Anyone who’s ever studied group theory will know that the result of rotation plus a translation can always be done using ONLY a rotation - it’s just a question of figuring out where the centre of rotation needs to be.

Since the shape we are using, a square, has order 4 rotational symmetry itself, there are in fact four possible centres of rotation, depending on which of the four sides of the tilted square eventually coincides with the base of the upright square. For instance, a 30 degree clockwise rotation around the top rotation point will align the squares such that the bottom right side of the tilted square becomes the base, whereas a 60 degree anti-clockwise rotation around the lowest rotation point will also align the squares, but now the bottom left side of the tilted square coincides with the base of the upright square.

As an aside it is interesting to note that these four points are collinear.

This week’s challenge is to find the co-ordinates of two of these four centres of rotation. Or for extra kudos, the coordinates of all four.