Forget computing the Kazhdan-Lusztig polynomials for E8, a far more important theorem has apparently been proved. I won't state the answer here as it's fun to think about the problem for yourself. But a number of proposed proofs of solutions to the Angel problem have been informally published, at least electronically.
Firstly, what is the problem? Wikipdia has a succinct write up. But here's a quick summary anyway:
The game of Angels and Devils, invented by John Conway (the same Conway as in the Game of Life, the Monstrous Moonshine conjectures, the proof that subatomic particles have free will and so on), is played on an infinite chessboard. Fix some constant k. One player plays the Angel. The other plays the Devil. On the Devil's turn, he may eat one square on the chessboard. (The devil doesn't actually have a piece, he just eats whatever square he wants.). On the Angel's turn she jumps to a square (distinct from the current one) up to k king's moves away. The Devil wins if the Angel has no legal move. The Angel wins if she has a strategy that guarantees always to have a legal move. For each k, who wins? Until recently it was unknown if the Angel could win even for arbitrarily large k. This may seem like a surprise, after all, if the angel can move 1,000 king's moves, surely she can easily evade the jaws of the devil who can only chomp one square at a time. But surprisingly, the obvious Angel strategies that come to mind can be countered.
What's nice about this problem is that it's so easy to state, and yet has stumped some of the smartest people who like to think about these problems.
Two papers with possible solutions are Máthé's and Kloster's. I don't think either of them have formally been published yet so I don't know if the peer review is complete.
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