Over at Neighborhood of Infinity, there's a mention that the Angels and Devil game might have been solved. Don't know what it is? It's a game played on an infinite checkboard. The Angel begins at the origin, and can move to any square that can be reached in k moves of a chess king. The Devil can then destroy a square anywhere. The Angel wins if he can evade the Devil. The Devil wins if he can box in the Angel. The game was proposed by Conway, and has up until now remained unknown as to whether there was a reasonable strategy for the Angel.
I haven't read them over in any detail, but it's the kind of neat recreational math stuff I like.
If you take the sequence of primes, and concatenate the first n of them together, how many of these are prime? It's not hard to see that 2, 23, and even 2357 is prime. The next term has 355 digits:
235711131719232931374143475359616771737983899710110\ 310710911312713113713914915115716316717317918119119\ 319719921122322722923323924125125726326927127728128\ 329330731131331733133734734935335936737337938338939\ 740140941942143143343944344945746146346747948749149\ 950350952152354154755756356957157758759359960160761\ 3617619631641643647653659661673677683691701709719
The concatenation of primes up until 1033 also appears to be prime.