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At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

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Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

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Take Ten

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3 cube?


Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem requires a lot of calculations of surface areas, within a rich problem solving context.

Possible approach

This printable worksheet may be useful: Cuboids.

Work with a specific cuboid, eg $2 \times 3 \times 5$, or a breakfast cereal box, to establish how to calculate surface area of cuboids. Students could practise working out surface area mentally on some small cuboids made of multilink cubes.

Present the problem, ask students to keep a record of things that they tried that didn't work (and what was wrong) as well as things that did work. In this initial working session, try to ensure that students are calculating surface area correctly. This spreadsheet may be useful (for teachers' eyes only!).

It may be appropriate to draw a ladder on the board, with this on the steps (starting from the bottom):
- calculations going wrong
- no solutions yet
- one solution
- some solutions
- all solutions
- why I am sure I have all the solutions
- I'll change the question to...

A short group discussion could suggest strategies to help students move on up the ladder, before they continue with the problem.

This might be a good lesson in which to allocate five minutes at the end to ask students to reflect on what they have achieved, which methods and ideas were most useful, and what aspects of the problem remain unanswered.

Key questions

  • Have you found none/one/some or all of the solutions
  • Is there a cube that will work?
  • How might you organise a systematic search for the cuboids with surface area $100$?

Possible extension

The main extension activity could focus on the convincing argument that all solutions have been found. Once this has been answered, you might like to consider these extensions:
  • Express the method for calculating surface area, algebraically.
  • What surface area values will generate lots of cuboids and which give none or just one?
  • Could you set up a spreadsheet to help with the calculations?

Possible support

In groups, or as a class, keep a record of all cuboids whose surface areas have been calculated.
Award ten points for a bulls eye "$100$", fivepoints for each $95-105$, and two points for $90-110$.
Any miscalculated results could lose points, providing motivation for peer checking, and helping each other.

A sheet showing a net of a cuboid, like this , may help students to organise their working and ideas.