Martin Gardner passed away yesterday, at 95; here is the NYTimes obituary.

I grew up with his puzzles and mathematical recreations, as I’m sure many mathematicians of my generation (and generations before) did. I am reminiscing this evening about interesting topics that Gardner introduced me to, and the following geometric gem comes to mind. The problem was apparently first proposed by Henri Lebesgue in 1914 and is still open, so it is almost 100 years old now.

Question: Find a universal cover of least area in the plane, meaning a set having a subset congruent to every planar set of unit diameter.

If one restricts to convex , then such a minimum is guaranteed to exist by the Blaschke selection theorem, a compactness result for sequences of bounded convex sets. A hexagon of unit width does the trick, but surprisingly, one can take off tiny pieces and still have a universal cover.

No one has ever exhibited such a minimum with proof, although there have been a few false claims of optimal solutions. I just searched and found an amusing 1992 Geometriae Dedicata article by H. C. Hansen in which the author shaves off the then reigning world record. I am not sure if anyone has improved on this result since, but this might be a fun thing to think about; indeed, Hansen suggests in the article how computers might help solve the problem, and computers have come a long way since 1992.

Gardner’s name came up in conversation just the other day. Tom Rokicki came to talk to my Rubik’s Cube class at Stanford about his work on “God’s algorithm” — Tom set a world record recently by proving that every Rubik’s Cube, no matter how scrambled, can be solved in 22 or fewer face turns. I asked him how long it would be until we have a proof of the conjectured answer of 20. He said he thought he might have 21 in the next few months, and then said that he wanted to have it down to 20 moves by Gathering for Gardner 2012. I hope that the Gatherings for Gardner continue, and it would be nice to see Tom succeed in his goal too.

I am not sure if anyone in history has ever made more people smile and shake their head about mathematical things than Martin Gardner. His writings illuminated it’s magical and mysterious qualities. Gardner was also famous as skeptic and a debunker, like many magicians before him. The NYTimes obituary reports:

He ultimately found no reason to believe in anything religious except a human desire to avoid “deep-seated despair.” So, he said, he believed in God.

Well godspeed to you Martin, and thanks for all the wonderful writings and beautiful thoughts.

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