All Wrapped Up

What is the largest cuboid you can wrap in an A3 sheet of paper?
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Well, we're often wrapping up presents - I find that many of them arrive in boxes that are cuboid. Remember those? A cuboid has 6 faces; each face a rectangle (including squares); each face paired with one the same on the opposite side and all the angles 90 degrees. So they might look like these:

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All Wrapped Up


Now let's talk about the wrapping paper. You are probably used to using A4 paper for all kinds of things. A4 paper looks a bit like this:

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All Wrapped Up


But two A4s together make A3:

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All Wrapped Up


So we have:

Image
All Wrapped Up


Which could look better as:

Image
All Wrapped Up


Now for easiness we'll approximate the size of the A3 sheet to 40cm by 30cm.

The wrapping paper we'll be using is A3-size, like the picture above.

So, we're wrapping up a cuboid in this A3 wrapping paper.

The challenge is to find the largest cuboid that can be covered completely with the A3 sheet - you will probably want to stick to whole numbers of centimetres!

Rather than wrapping the cuboid like we might do a birthday present, let's think about it as if we're trying to have as little overlap as possible.

You might do this practically or you may use diagrams or even a spreadsheet - it is up to you.