I was in anticipation all last week. Ernie, who had been travelling for several months, was finally coming home. So over the weekend I dropped in on him.
He was bursting with news. "Some wonderful duty-free deals at the moment!" he exclaimed. "Whiskey and room-temperature superconductors are on special" he continued, pointing to an enormous barrel in the corner of the room and a very large envelope resting on the kitchen table. Printed on the envelope were the words:
ACME INDUSTRIES 2x Room-temperature superconducting cubical boxes dimensions (HxLxW) = 1m x 1m x 1m (some assembly required)
"How can you fit boxes of any sort into an envelope?" I asked, somewhat confused.
Ernie opened the envelope and slid out the contents. "Quite easy if you flat-pack it", he replied. "The boxes come in the form of connected polyhedral nets. You just fold them up and then solder along the joins to make the cubes". But then his face fell - "Dang, there appears to be only one net inside the envelope".
I gathered up the discarded packaging and, with my fingernail, picked at the ACME label. Underneath it was a second label that read:
ACNE IMDUSTRIES 1x Room-temperature superconducting cubical box dimensions (HxLxW) = 1m x 1m x 1m (some assembly required)
"Gads!!", he exclaimed "another complete rip-off from Kzijekistan! I really need two boxes for my next experiment". He held a small device up against the material and a small light flickered green. "Well," he continued ruefully "at least the material itself is still a room-temperature superconductor, and the net is precisely large enough to fold into a 1m x 1m x 1m cubical box - so I guess I have only been half ripped-off'.
"Well," I ventured, "could you cut the net up to make a bunch of smaller pieces? Then you could fold and solder them to make two smaller cubes".
"Brilliant idea!", ernie replied "the experiment will still work if the boxes are a bit smaller - as long as I have two of them. But it is most important that each box is folded from a single piece of superconducting sheet because the solder joints wont be superconducting." He handed me a small pair of scissors, a soldering iron and s reel of solder. "Tell you what" he said "how about you do the cutting, folding and soldering while I go and find paper and a pencil so I can write a rude letter of complaint to the shop I bought this from?".
I was a bit nervous but eventually worked out how to cut two smaller polyhedral nets from the original (obviously there were some wasted bits at the edges), and assembled two identical cubical boxes. Ernie was most complimentary when he returned and measured the dimensions of the boxes i had just assembled. "Perfect" he said, "the boxes are identical cubes and if I am not mistaken, they are the largest that could possibly be constructed from any polyhedral net for a 1m x 1m x 1m cubical box". He offered to reward me with an amount of vintage whisky equal in volume to that of one of the cubes I had constructed. "Just bring along a large enough container to hold it all and I will decant your share from the barrel."
Wel, I do love whiskey, and want to ensure I get my full reward. But I don't want to appear greedy - it would be embarassing to turn up with a container that was too large. Can anyone help me work out the volume of one of the smaller cubical boxes?
To avoid any confusion
- The superconducting sheet is very thin (so its volume can be ignored).
- It can be folded along any line - just like a piece of paper.
- The original (unassembled) cubical box and the two identical smaller cubical boxes each have six faces.
- The original net was just big enough to produce a 1m x 1m x 1m cube - no overlaps at any point.
- The original polyhedral net and each of the two smaller nets consisted a single continuous piece before folding.
- Both the original net, and the replacement nets, can be soldered together along edges and/or faces to form the cubes.