Hot water CAN freeze more quickly than cold water

I heard that statement years ago and dismissed it, but the subject was brought up recently in a Reddit thread, which linked to an extended explanation by a physicist.  Herewith brief excerpts:

Hot water can in fact freeze faster than cold water for a wide range of experimental conditions.  This phenomenon is extremely counterintuitive, and surprising even to most scientists, but it is in fact real.  It has been seen and studied in numerous experiments.  While this phenomenon has been known for centuries, and was described by Aristotle, Bacon, and Descartes, it was not introduced to the modern scientific community until 1969, by a Tanzanian high school pupil named Mpemba...

This seems impossible, right?  Many sharp readers may have already come up with a common proof that the Mpemba effect is impossible.  The proof usually goes something like this.  Say that the initially cooler water starts at 30°C and takes 10 minutes to freeze, while the initially warmer water starts out at 70°C.  Now the initially warmer water has to spend some time cooling to get to get down to 30°C, and after that, it's going to take 10 more minutes to freeze.  So since the initially warmer water has to do everything that the initially cooler water has to do, plus a little more, it will take at least a little longer, right?  What can be wrong with this proof?

What's wrong with this proof is that it implicitly assumes that the water is characterized solely by a single number — its average temperature.  But if other factors besides the average temperature are important, then when the initially warmer water has cooled to an average temperature of 30°C, it may look very different than the initially cooler water (at a uniform 30°C) did at the start.  Why?  Because the water may have changed when it cooled down from a uniform 70°C to an average 30°C.  It could have less mass, less dissolved gas, or convection currents producing a non-uniform temperature distribution.  Or it could have changed the environment around the container in the refrigerator.  All four of these changes are conceivably important, and each will be considered separately below...

Prepare to spend some time at the link if you want to understand the details.  I'll accept it on faith now.