which group wins?

I think Group B can solve the problem quicker.
I guess people tend to be more excited or got enthusiasm when there's a reward for it.
And this type of problem needs people to be cool and calm to figure it out.
Group A has too much pressure to figure this problem out and they can't be cool.
Or some of them might be given it up before starting to solve.
If you think you can't get reward for it because the rivals are too many, why do you have to do this? Isn't is just wasting the time?
Then you might not to be able to get satisfaction by figuring it out after the all rewards were taken by the others.
On the other hand, Group B does't need to be too eager to solve it, so they can try to figure it out like a cross word on the paper calmly and easily.
Maybe their movement would be slower, but everybody has motivation like compliting a game.
It's good to have many motivated people to figure the problem, I suppose.
So I think Groupe B will solve this much more quicker.

what motivates people?

Sorry, about a candle thing, I really hate thinking about those stuff. So let me skip that part.

So, what motivates people? I guess it really depends on people.
But personally, get rewarding from someone really motivates me only if the task is simple and everybody can do it with a little effort.
If a task is really hard to get a reward, I won't even try. I will give up before I give it a shot.
The bottom line, we are not equal. People are not same. There are very smart people and some of them are really not. So I think what should be fair to everyone is tasks.
Rewards sometimes works for me, but not all the time.