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?

Answer 1

5

Answer 2

The area of each rectangle is 6, so the area of the square must be divisible by 6. The smallest square side length that works is 6. It is easy to see that we can tile a 6 by 6 square with 2 by 3 rectangles, then you can split the rows into pairs of two, then cover each pair with two rectangles laid end-to-end. Since the area of the square is 36, and each rectangle has area 6, the number of rectangles is 6.