Brainteaser 1631: Ali Baba’s boxes

From The Sunday Times, 12th December 1993 [link]

The royal jewellers Fabulé make standard-sized golden cubic boxes which are diamond-encrusted on the outside. They make these from large sheets of gold which have been encrusted with diamonds on one side and they then cut out the required shapes to make the boxes.

For example, the 3-by-4 cruciform shown on the left below provides a shape which folds up into a cubic box, but the strip on the right does not.

Recently thieves broke into the Fabulé workshops and stole various cut-out pieces of the diamond-encrusted sheeting. They did not realise that in fact they had taken the mis-cuts: all the pieces that consisted of six squares but none would actually fold into a box. And their haul consisted of one piece of each possible faulty shape.

How many pieces did they steal?

This puzzle was selected for the book Brainteasers (2002, edited by Victor Bryant). The text was changed, but the substance of the puzzle remains the same.