How do I prove that it is possible to cover the whole plane with squares with side 1?
Please suggest different approaches to the above problem.
Thank you.
How do I prove that it is possible to cover the whole plane with squares with side 1?
Please suggest different approaches to the above problem.
Thank you.
If you aren't concerned about overlap (which you don't mention in your statement), then for every point x in the plane, include a square of side length 1 whose lower lefthand corner is x. Not only is everything covered, but each point gets its own personal square.
If you'd like a slightly less silly cover, then anchor the lower lefthand corner of a square at each point with integer coordinates. If you remove the top side and right side of your square, then this scheme will cover the entire plane without overlap.