Jack and the Box Company

[Problem by Stephanie Wilcox. ASCII art and extensions by Sam Rebelsky]

Jack owns his own box company and he sure likes making boxes. However, ever since his grandfather owned the box company, the boxes looked like this before assembly:

I lost the picture. Is this one okay?
  +---+
  |   |
  +---+---+---+---+
  |   |   |   |   |
  +---+---+---+---+
  |   |
  +---+

Since Jack is an ambitious box maker, he wants to know if there are other ways to make boxes. Can all combinations of six squares be assembled to make boxes? Are there shapes that are easier to make into boxes than the figure that Jack already uses?

Given that the boxes are cut from larger pieces of cardboard, is there an arrangement that is better or worse from the "ease of cutting" perspective or that ensures better or worse use of materials. Does your answer depend on the size of the cardboard relative to the size of the squares?


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