The Mpemba effect
Hot water freezes faster than cold water. Or does it? Cold water at 0.01 ºC will probably freeze before water at 99.99 ºC, and a small drop of cold water will probably freeze faster than a large quantity of hot water. So it’s certainly not true that hot water always freezes faster than cold water. It is true in some cases, however, and there are enough of these cases to make the property interesting.
The fact that hot water sometimes freezes faster than cold water has been named the "Mpemba effect" after Erasto Mpemba, who brought it to the attention of the scientific community in 1963 while he was a high-school student in Tanzania. It turns out that stating the Mpemba effect in a careful way that’s possible to verify experimentally is actually fairly tricky. You probably need something like this: "There exists a set of physical parameters and a pair of temperatures such that given two samples of water identical in these parameters and differing only in their initial uniform temperatures, the hotter of the two will freeze sooner."
If that’s what it takes to say it carefully, it’s easy to see why "hot water freezes faster than cold water" is preferred by most people.
You see the same preference for shorter yet not-quite-as-accurate descriptions of cryptographic schemes. It’s easy to describe the Diffie-Hellman key exchange like this:
- Alice uses her private key a to calculate ga, which she sends to Bob
- Bob uses his private key b to calculate gb, which he sends to Alice
- Alice calculates the shared secret gab as (gb)a
- Bob calculates the shared secret gab as (ga)b
This is as sloppy as saying that hot water freezes faster than cold water, but it’s also good enough for almost every time that you need to describe the Diffie-Hellman key exchange.
If you take the time to say everything precisely that you’ll find that it takes too much long to say anything of consequence. So we accept a certain amount of inaccuracy in day-to-day conversation and reserve being careful to things that we put in writing. Try to describe something as simple as the Diffie-Hellman key exchange in careful and precise language and you’ll find exactly how hard this can be.