Paul Erdős, a great Hungarian mathematician of the 20th century was famous of proposing cash prizes for his math problems. He never owned anything (everything he had could be put into a small suitcase). He thought possession is evil. He gave away his money to poor students and for his cash prizes pay-out. There were only two things he was addicted to: math and meth (with prescription, after his wife passed he suffered from depression). Worried about his addiction to meth, Ron Graham, Erdős’ colleague and boss at AT&T Bell Lab, offered him \$500 if Erdős can stop taking meth for a month. Erdős accepted the offer and he actually managed to stop taking meth for a month. After getting \$500 from Graham, he started taking meth again. Erdős reportedly said something like this: “I couldn’t do math during that time. Every time I looked at a piece of paper, nothing came out of it.”
There is another mathematician who proposed cash prizes for his math problems. His name is Michel Talagrand. He is a well known French mathematician in probability. His rule is funny. Only the first person who solves gets the prize. You don’t have to be the one who solved the problem to claim the prize. You can still receive the prize if you purchased solution, but at half the prize money. Also he asks to submit solutions before he becomes senile because he won’t pay if he can’t understand solution :). The largest prize is \$5000 one. Unfortunately, someone (actually two people) solved it and they together received the money. There are three more problems left. Each has \$1000 prize.
Mathematicians don’t tackle difficult problems for money or fame. But some cash prizes can surely add a little more fun and excitement. Although not a cash prize, Arthur Besse, a pseudonym for a group of French differential geometers, offers a nice meal at a Michelin-starred restaurant in exchange for a new example of compact Ricci-flat manifolds (they are very difficult to find).
I like “real and good” mathematicians. (Regrettably, I am not one of them. I am not a real mathematician because I am not working on serious mathematics problems. Of course, if one is not a real mathematician, he can’t be a good mathematician.) They are not just smart but humble (No offense but you can hardly find someone among “real and good” mathematicians like those arrogant a$$holes, who act like they know everything, among hep-th people. Make no mistake, there are also so many hep-th people whom I respect and admire greatly.), humorous, have interesting personal characters, and certainly know how to have fun.
Are you up for going after math bounties? Go at them!