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I produced a diagram, and concluded that the net torque about the bottom of the board is 0. The torque forces I included were the torque from the centre of gravity, the torque from the wall, and the torque induced by friction on the wall. I went through and ended up with a trigonometric equation in two variables, theta and mass, that I could not cancel. I am not even going to bother to show my work because my logic must be wrong.

Any suggestions?