To cover a square region exactly, the side length of the square must be a common multiple of 2 and 3, the dimensions of the rectangles. The smallest common multiple of 2 and 3 is 6, so the side length of the square must be 6 units. This means the area of the square is 6^2 = 36 square units. The area of each rectangle is 2 times 3 = 6 square units. To cover the square completely, I need to use 36 divided by 6 = 6 rectangles. This is the smallest possible number of rectangles, because if I use fewer than 6, there will be gaps or over-hangs. 
What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?