When did the snow start to fall?

Solution: Assume that the snow was falling at a constant rate of $r$ inches per hour, and the snow plower can remove snow from the streets at a constant rate. The width of the snow plower is $w$ inches. Assume the snow starts to fall $x$ hours before 12 noon. Therefore, we have the following observations:

At 12 o’clock, $x\times r$ inches of snow on the unplowed streets

At 13 o’clock, $(x+1)\times r$ inches of snow on the unplowed streets

At 14 o’clock, $(x+2)\times r$ inches of snow on the unplowed streets

The total volume of snow removed in the first hour would be:

$$6\times w\times\dfrac{x\times r + (x+1)\times r}{2}\ \ \ \ \ {miles\times inch^2}$$

The total volume of snow removed in the second hour would be:

$$3\times w\times\dfrac{(x+1)\times r + (x+2)\times r}{2}\ \ \ \ \ {miles\times inch^2}$$

Because the snow plower removes snow at a constant rate, we have:

$$6\times w\times\dfrac{x\times r + (x+1)\times r}{2}=3\times w\times\dfrac{(x+1)\times r + (x+2)\times r}{2}$$

Solving the above equation, we have $x=\dfrac{1}{2}$. Therefore the snow started to fall at $\boxed{11:30 AM}$.