A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?
I need to find the possible ways to divide the plywood into $4$ congruent rectangles. One way is to cut it along the middle of the length and the width, forming $4$ rectangles that are each $2$ feet by $4$ feet. The perimeter of each rectangle is $2 + 4 + 2 + 4 = 12$ feet. Another way is to cut it along the middle of the length and then cut one of the halves into two equal parts along the width, forming $4$ rectangles that are each $1$ foot by $8$ feet. The perimeter of each rectangle is $1 + 8 + 1 + 8 = 18$ feet. A third way is to cut it along the middle of the width and then cut one of the halves into two equal parts along the length, forming $4$ rectangles that are each $4$ feet by $2$ feet. The perimeter of each rectangle is the same as the first way, $12$ feet. I can't think of any other ways to divide the plywood into $4$ congruent rectangles, since any other cut would either leave some wood left over or make the rectangles not congruent. Therefore, the greatest possible perimeter of a single piece is $18$ feet and the least possible perimeter of a single piece is $12$ feet. The positive difference between them is $18 - 12 = 6$ feet. 