I think it's best if you keep the game running until there are prizes left on it to be claimed.
Sounds like you are suggesting the game to never expire unless all prizes are claimed. That is my first idea as well, but then I realized it won't work well unless the site is very popular.
Take image #4 as an example, it is up for more than 2 days now, and only 121 squares out of 551 has been clicked. So it would probably take 10 days to get it all clicked. Or in other words, the players will need to wait for 10 days to finally know the unhashed prize string and to verify their bets. IMHO, that kind of wait is too long, and thus I think the current expiry is good.
