Somebody (who didn't give their name) said that they thought the best place to start was a little over half the length, height and width of the box, which seems very sensible.
A number of you who have tried it drew tables to show possible lengths, heights and widths of the cuboid with the volume for each. This also seems like a good idea. Don't forget that having some cubes to hand might also be very helpful, or if you're good at 3D drawing, you could sketch out the cuboid in the box to check whether another would fit.
Well done to Joe, Mark, Lucy and Alex from Hazel Grove High School, Stockport who said that the smallest cuboid they could find is 7cm by 7cm by 7cm. They wrote:We used the method of "slightly larger than halves".
They really persevered with their solutions. Fantastic work.
There is an alternative solution which you may want to think about too. Have a look at the hints and let us know if you find anything out!