The Polynomial Game
This puzzle is a generalization of a problem from the 1977 USSR math Olympiad:
At the beginning of the game you are given a polynomial, which has 1 as its leading coefficient and 1 as its constant term. Two people play. On your turn you assign a real value to one of the unknown coefficients. The person that goes last wins if the polynomial has no real roots at the end. Who wins?
It is clear that if the last person’s goal is for the polynomial to have a root, then the game is trivial: in this case, he can always make 1 a root with the last move. Also, an odd degree polynomial always has a real root. Therefore, to make the game interesting we should assume that the degree of the polynomial is even.
Though I can’t imaging myself ever being interested in playing this game, figuring out the strategy is a lot of fun.
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Misha:
Especially for children coming to grips with mathematics.
12 October 2008, 6:28 am