Boxed Blocks
THE CHALLENGE
There is a wooden box, without a lid, holding a 4 x 4 x 4 collection of 64 blocks. How many of the blocks touch some part of the box?
EXPLORATION
What happens for a 5 x 5 x 5 collection of 125 blocks in a bigger box? What about other sizes of boxes? How do these answers change if the boxes have a lid that touches the top row of blocks?
Notes
THE CHALLENGE
There are different ways to go about solving this problem.
- Think of the 2 x 2 x 3 = 12 core of blocks that don’t touch the sides.
- If it had a lid, there would be 2 x 2 x 2 = 8 blocks not touching the box. In addition to those 8, there would be 2 x 2 x 1 = 4 blocks only touching the lid.
- You can add up the blocks along the sides and subtract them from the 4 x 4 x 4 = 64 total. This is the hardest way. There are 4 x 4 x 1 = 16 blocks on the bottom of the box. There are 3 x 12 = 36 blocks around the sides that don’t touch the bottom. Therefore, there are 64 – (16 + 36) = 12 blocks on the inside.
EXPLORATION
Look at these only using the first method.5 x 5 x 5. Without a lid, there are 3 x 3 x 4 = 36 blocks not touching the box. With a lid, there are 3 x 3 x 3 = 27 blocks not touching.
a x b x c, where c is the height. Assume these numbers are each at least three. Without a lid, a and b are reduced by 2 and c by 1, so there are (a – 2) x (b – 2) x (c – 1) blocks. With a lid, (a – 2) x (b – 2) x (c – 2) blocks not touching.