You are on an infinite Cartesian plane at the origin (0,0). For every integer pair of coordinates (n,m) there's a null-dimensional point (that is, the point has zero width and height).

Some of these are "visible" to you, but some others are "invisible". For example, the point (2,2) is not visible from the origin since it is "blocked" by (1,1). On the other hand, (3,5) is "visible" to you since there are no other points in the way.

Where can you build an "invisible" unit (1x1) square (all four vertices of which are "invisible" points) as near as possible to you - and the origin?

I won't post a full solution now, but there is one useful point that could be made. For any point (x, y), the point will only be invisible if x and y share a common factor. For example, if they both have a factor of some number n, such that x=an and y=bn, then the point (x, y)=(an, bn) will always be invisible because the point (a, b) will be in the way.

In other words, if the two coordinates of a point share any common factor, there will be some point directly between the given point and the origin, found by factoring out the common term from both values.

Posted by DJ on 2003-08-24 07:08:25