Rectangles

Take rectangles with the following dimensions (width × height):

• 4 × 3
• 2 × 3
• 3 × 3
• 3 × 3
• 2 × 1
• 2 × 5

If you put them next to each other in a line the smallest rectangle that bounds them is of area (4 + 2 + 3 + 3 + 2 + 2) × 5 = 80. If you stack them on top of each other then the smallest rectangle that bounds them is (3 + 3 + 3 + 3 + 1 + 5) × 4 = 72.

However, if you arrange them a bit more carefully it is possible to arrange them into a rectangle of area 48¹. How?

What is the smallest bounding rectangle that covers the following:

• 8 × 10
• 5 × 10
• 7 × 8
• 2 × 4
• 6 × 3
• 6 × 5
• 7 × 6
• 3 × 4
• 10 × 7
• 2 × 8
• 2 × 10
• 5 × 8
• 1 × 2
• 1 × 2

If you can work out that, how about these forty rectangles?

5 × 9, 5 × 8, 6 × 3, 5 × 10, 2 × 6, 9 × 6, 7 × 6, 4 × 6, 1 × 8, 3 × 3, 8 × 2, 10 × 7, 5 × 1, 7 × 2, 9 × 9, 5 × 3, 3 × 10, 9 × 6, 7 × 1, 8 × 9, 8 × 2, 6 × 7, 1 × 2, 10 × 7, 8 × 10, 7 × 3, 9 × 8, 7 × 3, 10 × 3, 3 × 6, 5 × 2, 6 × 9, 1 × 6, 7 × 8, 3 × 8, 8 × 9, 9 × 1, 4 × 5, 9 × 10

Do the answers come out differently if you are allowed to rotate the rectangles?