Both the early scientific history of this effect, and the story of Mpemba's rediscovery of it, are interesting in their own right -- Mpemba's story in particular provides a dramatic parable against making snap judgements about what is impossible. This is described separately below.
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 -- the 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.
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.
Why hasn't modern science answered this seemingly simple question about cooling water? The main problem is that the time it takes water to freeze is highly sensitive to a number of details in the experimental set- up, such as the shape and size of the container, the shape and size of the refrigeration unit, the gas and impurity content of the water, how the time of freezing is defined, and so on. Because of this sensitivity, while experiments have generally agreed that the Mpemba effect occurs, they disagree over the conditions under which it occurs, and thus about why it occurs. As Firth wrote "There is a wealth of experimental variation in the problem so that any laboratory undertaking such investigations is guaranteed different results from all others."
So with the limited number of experiments done, often under very different conditions, none of the proposed mechanisms can be confidently proclaimed as "the" mechanism. Above we described four ways in which the initially warmer water could have changed upon cooling to the initial temperature of the initially cooler water. What follows below is a short description of the four related mechanisms that have been suggested to explain the Mpemba effect. More ambitious readers can follow the links to more complete explanations of the mechanisms, as well as counter- arguments and experiments that the mechanisms cannot explain. It seems likely that there is no one mechanism that explains the Mpemba effect for all circumstances, but that different mechanisms are important under different conditions.
SurroundingsA final difference between the cooling of the two containers relates not to the water itself, but to the surrounding environment. The initially warmer water may change the environment around it in some complex fashion, and thus affect the cooling process. For example, if the container is sitting on a layer of frost which conducts heat poorly, the hot water may melt that layer of frost, and thus establish a better cooling system in the long run. Obviously explanations like this are not very general, since most experiments are not done with containers sitting on layers of frost.