interesting puzzle

In addition to its weekly mathematics puzzles, Le Monde is now publishing a series of vulgarisation books on mathematics, under the patronage of Cédric Villani. Jean-Michel Marin brought me two from the series, one on the golden number and one on Pythagoras’ theorem. (This is actually a translation of a series published by El Pais last year.) Those books are a bit stretched given the topic, even though I enjoyed the golden number (the second one having a lot of redundancy with the first one.) However, I came upon an interesting question, namely about the maximum size of a cube that could fit through a tunnel drilled through the unit cube. Sadly, I could not find an answer to this problem on the web, even though the book mentions a solution with a side larger than one…