Train
Lifer
But there's no benefit. You're burning processing power for a random reason to generate questionable income. Why not put that to good use and have a paid distributed computing model where your processing power is used to actually do something useful...
If it were possible to integrate such a problem into the mining of bitcoins, I am sure it would be done. However bitcoin was setup so that the answers are unknown (and can't be known until previous blocks are uncovered) or else someone would just "discover" all possible blocks at once.
But don't worry, once all possible blocks are found, mining will stop forever. Until someone invents a new digital currency.