There are five elementary problems on this list.  Knowing Conway, the $1000 reward will probably come in a box that you have to spin and tap appropriately to open...
[...] John Conway is offering $1,000 for solutions (either positive or negative) to any of the  following problems. If you solve one of these, you can reach him by sending snail mail  (only) in care of the Department of Mathematics at Princeton University.

Problem 1.  ‘Sylver’ coinage game (named after Sylvester, who proved it  terminates):

The game in which the players alternately name positive integers that are not sums of  previously named integers (with repetitions being allowed). The person who names 1  (so ending the game) is the loser. The question is: If player 1 names ‘16’, and both players play optimally thereafter, then  who wins? [...]
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