Thursday, November 12, 2015

Probabiliy: Who has the winning strategy?

Consider a polynomial

Nigel and Jessica are playing the following game. In turn, they choose one of the coefficients $a_0,\,\cdots,\,a_{2011}$ and assign a real value to it. Nigel has the first move. Once a value is assigned to a coefficient, it cannot be changed any more. The game ends after all the coefficients have been assigned values.

Jessica's goal is to make $P(x)$ divisible by a fixed polynomial $m(x)$ and Nigel's goal is to prevent this.

Which of the players has a winning strategy if $m(x)=x-2012$?

Solution provided by Mark, another contributor from this blog:

In order to win, Jessica wants to have at the end (since she has the last move):






No matter what values Nigel and Jessica have chosen for the first 2,011 coefficients, all Jessica has to do is choose for the last ($a_\ell$ where $0\le\ell\le2011$) which can be obtained from:


Solving for $a_{\ell}$, we find:


With this value for $a_{\ell}$, Jessica is assured of winning the game.


  1. Beautiful explanation on this problem Isabelle. Well done my friend.(Y)

  2. This beautiful explanation belongs to Mark! :P