Making boxes

This is an investigation involving the volume formula. If we allowed non-integer sized cuts, would this be a quadratic optimization problem? What if we allowed rectangular cuts?


The original problem is posed here. I have fixed one ambiguous phrase, and amended one confusing directive.


Start with some sheets of squared paper measuring 15×15 cm, and use them to make little boxes without lids.

You do this by cutting out squares at the corners and then folding up the sides. (The folds are indicated by the dotted lines in the diagram.)

Begin by cutting a 1×1 square out of each corner. Fold up the sides. What is the size of the base? How high are the sides? So what is its volume?

Now cut a 2×2 square out of each corner and fold up the sides. Does it look as if it holds more than the first box, less than the first box or just the same amount? What is the size of the base now? How high are the sides now? So what is its volume?

Now cut a 3×3 square out of each corner and fold up the sides. Does it look as if it holds more than the other boxes, less than the other boxes or just the same amount? What is the size of the base now? How high is it now? So what is its volume?

If you keep on doing this, taking larger and larger squares from the corners, which box will have the largest volume?

What if you had started with a different sized square? What would be the optimal size corner to remove in that case? Can you generalize to n×n? What if you had started with some rectangle, say 15×20? Can you generalize to mxn?

Here's the solution for the 15×15 cm piece.