Robots

There are two possible ways for the red robot to move. It could go (1,1), (1,2), (1, 3), (2, 3), (3, 3), (3, 2), (3, 1), (2, 1) and back to (1,1). We’ll deal with the other case in a minute.

Similarly there are two ways for the blue robot to move. Suppose that it goes, go (4,4), (4,3), (4, 2), (3, 2), (2, 2), (2, 3), (2, 4), (3, 4) and back to (4,4). If you compare the different grid references you will see that the two robots are never on the same square at the same time.

But the blue robot might go (4,4), (3,4), (2, 4), (2, 3), (2, 2), (3, 2), (4, 2), (4, 3) and back to (4,4). In this case the two robots are on the same square at the same time twice – the squares (2,3) and (3,2).

If the red robot goes round its square the other way (anti-clockwise) it will meet up twice with the blue robot if the blue robot goes clockwise. Then they meet on the squares (3,2) and (2,3). If the blue robot insists on going anticlockwise, then they won’t meet up at all.

So the two robots can be made to simultaneously occupy the squares (2,3) and (3,2).