In my opinion, both Group A and B figure out the problem at about the same time.
Even though I'd be happy if I could get a prize, what I can solve is possible to solve, and what I can't is can't... I have this opinion because I usually do something not for getting the prize, but because I have to do it, or it is fun thing to do. Other than that, I mostly give up.
Talking about the motivation, Group A might be much more motivated than B because of the reward. Therefore, Group A might have a strength to think about the solution strategy longer. What I want to say is that the finishing time could be the same, but the motivation of Group A could be higher.
Just think of anything you really really want to buy or get right now. What if you can get it? Would you try? (By the way, my prize gonna be "a home theater set," or "Xbox" although they are very expensive...) I ask those questions to myself when I think about this topic. I'm not sure if I can figure out the problem, but I might try to do anything if I can get what i want, even if it's very difficult.
This is my rationale.
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