This problem is a great example of a "low threshold high ceiling" task that can be approached in different ways. As well as having an opportunity to practise visualising, 2D representations of 3D shapes, and volume calculations, students can be introduced to the idea of finding an optimal solution, a key idea in mathematics. As there are many ways to approach and solve the problem, students can be drawn into new ways of reasoning mathematically, using numerical, algebraic and graphical representations, and for the most curious student, it could provide a route into thinking about calculus!
This printable worksheet may be useful: Cuboid Challenge
"Imagine you have a 20 by 20 square sheet of paper. Now imagine cutting out a square of side 5cm from each corner, and folding up the flaps. What would the dimensions of the resulting box be? Check with your neighbour and see if you agree."
"What would the volume of the box be?"
Bring the class together and confirm the dimensions, and the resulting volume. Take time to discuss how students worked it out, perhaps using diagrams showing the net and the finished box.
"I wonder if we could make a box with a larger volume..."
Allow some time for the students to work in pairs to explore the effect of cutting out different squares on the volume of the resulting box.
Then collect the results so far in a table on the board or a spreadsheet.
"What's the maximum volume we've found so far? Can we do any better?"
Invite students to comment on patterns they can see in the results, and to make suggestions of methods that might lead to the optimum solution. This could include:
Encourage some students to try each method so that you can bring the class together to compare and discuss their results.
Once students have found the maximum for a 20 by 20 square, set different groups of students a different size of starting square. Can they use one of the methods to find the cut out that gives the maximum volume?
Finally, collect in a table the cut out that maximises the volume for each square, and invite students to look for a general pattern that will work for any size of square.
How can you be sure you have found the maximum volume?
Offer students 20 x 20 square grids and encourage them to make different sized boxes, working systematically and recording their results as they work.
Instead of starting with square sheets of paper, students may investigate rectangular ones. In order to make pattern spotting easier, you may wish to organise this in some way, for example giving different groups of students sets of rectangles (such as rectangles where the length is twice the width, three times the width, four times the width, etc.)
Students could use algebra to represent the relationships they find, and if they know some calculus, use that to maximise the volume.