1)

2)

3)

Multiply the first equation by 2, then add the equations.

We have .

Now, plug p in any equation and find q.

4) Plug x=4.2 and x=4.3 in equation and probably you'll obtain one negative value and one positive value.

5) Let x be the number of middle pieces.

The probability to pick one is .

On the other hand, the same probability is 1-0.008-0.164.

Equalize the two values and solve for x.