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“Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries?”

One hundred years ago, with those inviting thoughts, the German mathematician David Hilbert opened his landmark address to the International Congress of Mathematicians in Paris, laying out 23 of the great unsolved problems of the day. “For the close of a great epoch,” Hilbert declared, “not only invites us to look back into the past but also directs our thoughts to the unknown future.”

With another century ending — a whole millennium in fact — the pressure is all the greater to tabulate human ignorance with lists of the most enticing cosmic mysteries.

In May, the Clay Mathematics Institute of Cambridge, Mass., emulated Hilbert, announcing (in Paris, for full effect) seven “Millennium Prize Problems,” each with a bounty of $1 million.  The list is at: http://www.claymath.org/prize_problems/ .

And last month physicists, with a typically lighter touch, ended a conference on superstring theory at the University of Michigan with a session called “Millennium Madness,” choosing 10 of the most perplexing problems in their field. It was like a desert island game, involving some of science’s smartest people.

“The way I thought about this challenge was to imagine what question I would ask if I woke up from a coma 100 years from now,” said Dr. David Gross, a theoretical physicist at the University of California at Santa Barbara, as he unveiled the winners. He and the other judges made the selection, he noted, “in the middle and after this party in which we were sufficiently drunk.”

After weeding out unanswerable questions (like “How do you get tenure?”), the judges came up with enough puzzles to occupy physicists for the next century or so. There are no monetary prizes, though solving any one of these would almost guarantee a trip to Stockholm.