Enigma 1011: The ribbon’s reach
From New Scientist #2165, 19th December 1998 [link]
Mary is wrapping her last Christmas present, which is a rectangular box which measures 1 metre by 1 metre by 2 metres. She has attached one end of a piece of ribbon to a corner of the box. Amazingly, she finds that the ribbon is just long enough to reach any point on the surface of the box; however if it were any shorter it would not be able to do that.
How long, to the nearest millimetre, is the ribbon?