Without doubt, one of the largest differences between the USA and Israel is the microwave ovens. Whereas almost all microwaves I encountered in Israel had either analog rotary timers or preset “30 sec or 1 min” buttons, here in the states there is an overwhelming prevalence (100% of an astounding three cases) of numpad microwaves.

Seemingly, all you have to do is put in the number of minutes / seconds you want to heat, and fin, you are done.

But wait; is it minutes, or seconds? What happens if I put in a three or four digit number? Do I have to be an arithmetic expert to operate my microwave?

In what can only be stated as the “non-continuity of microwave space-time”, the input parsing is simple: if you put in xx:yy, it will run for xx minutes, and yy seconds. Simple and intuitive. The thing is, nothing constrains yy to be smaller than 60. 1:99 is as valid input as any, and will indeed run for 1 minute and 99 seconds (=159 seconds total). 2:00 is also a valid input, running for 2 minutes, 0 seconds (=120 seconds total).

This is the natural way to handle user input, and I totally approve of it, if only for the programmers’ and designers’ sake for not handling annoying details. There is a nice time discontinuity when you plot the actual cooking time against the numbers you punch in, if you arrange them in lexicographical order:

Starting from 60 seconds cook time, the user has **two** choices of how she wants the input to be shaped, rather than just the feeble one available with a rotary timer. This is in agreement with the USA’s enhanced economic and political freedom; it is no wonder that these microwaves are more prevalent here (as for me, you can find me standing dumbstruck in front of the machines, trying to decide which number I should punch in).

As the title of the above plot suggests, it is interesting to see how different minute lengths affect our options. The shorter the minute, the more overlap there will be, and the more options you will have, until finally, for SECONDS_PER_MINUTE = 1, we have 100(!) different options of input. Here is the example for a 30 second minute:

On the other hand, given that we work in base 10 and that our two digit numbers only go up to 99, if we had a longer minute (and kept the input method the same, allowing only two “seconds” digits), we would have gaps:

Not every desired time can be reached; we will likely not be seeing any 200 second minutes in the Imperial system any time soon.

This whole ordeal reminded me of a wonderful fact I stumbled upon that has to do with discretizing age. Consider the standard high school algebra question: Albert is X years old, and his son Jorgenhausfer is Y years old. When will Albert be twice as old as his son?

The question is easy, but one can also ask for *how long* Albert is twice as old as his son. It turns out that Albert will be twice as old as Jorgenhausfer for exactly one year, but that time period may be split into two sections, depending on their birthdays! I can do no better justice to the issue than the discussion given here:

http://www.math.brown.edu/~banchoff/twiceasold/