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Cuboid-in-a-box

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
We have a really good box to store things in.
It's a cuboid that's 12 cm long, 8 cm wide and 10 cm high.

cube on own
We so often get asked for the biggest something or other, so for a change, here's a challenge with a difference:

What is the SMALLEST CUBOID that you can put in the box so that you cannot fit another that's the same into it?

You could use little cubes that are 1 cm wide, long and high that clip together or make a cuboid that must be a whole number of cm wide, long and high.

Let us know what you come up with - and what is more important - how you came to get a solution.