A square ink stamp has sides of length 1 cm.
It is covered in black ink and carefully placed in the middle of a piece of white paper.
The stamp is then rotated $180^o$ about one of its corners so that all of the square remains in contact with the paper throughout the turn, leaving a black trail.
The stamp is then removed from the paper.
What area of paper, in cm$^2$, is coloured black?
This problem is taken from the UKMT Mathematical Challenges