[Ether]

See notes in red below:

Originally Posted by Ninja_Bait

I know I'm doing something wrong.

1. The wheel applies a force of Tau/R in the y direction. the component of the floor's reaction force on the wheel in the Y direction is tau/r There is no motion in the wheel so Fy must equal Tau/R. correct
2. There is no motion of the robot, so there must be a torque to resist that generated by Fy, namely that generated by Fx. Fx and Fy are at the same distance from the CoM and at opposite angles to the moment arm, so they must be equal in magnitude. correct
3. F is the magnitude of Fy added to Fx: I assume you mean vector addition. Tau*sqrt(2)/R correct

Here's where you go off the rails:
4. To rotate, Fy must exceed Fx. This yields a net torque in the clockwise direction, which is what I assume is the goal.
5. Fy>Fx
6. Tau/R>muN
7. Tau>muNR

I have a problem with my answer for B because I feel like Tau/R should not exceed the maximum static friction, because then there would be a lot of wheel spin, which doesn't click for me. I also realize it's what quinxorin already posted. I just don't know how to do it otherwise.