t-Mobile has these tiered refill cards for their prepaid mobile phones. The pricing table is here and reproduced below:

$10 for 30 minutes, expires in 90 days
$25 for 130 minutes, expires in 90 days
$40 for 208 minutes, expires in 90 days
$50 for 400 minutes, expires in 90 days
$100 for 1000 minutes, expires in 365 days

So which card should you buy? You could calculate a per minute cost and conclude that $100 for 1000 minutes is the most economical (plus it doesn’t expire for the longest time). Wrong!

It depends on how much you use the phone. The fact that the minutes expire makes the prepaid plan a virtual monthly plan in the regime where you do not use 1000 or more minutes per year, which is highly likely for people who choose prepaid phones to begin with (e.g. temporary visitors, odd occasions, emergencies, etc.). The constraint in that case is the expiration, not the number of minutes. If you blindly purchased $100 refills one after another, you’d have more and more unused minutes piling up. Sure, you could still use them, but even at $0.10/min. it is expensive compared to a straight monthly plan if you really mean to call that much. Of course you don’t, so now what?
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The great feature of an analog microwave oven controller is that setting a time is highly efficient and the effort taken to set the time (at worst) scales linearly with the heating need, which seems like a correct scaling. It has additional features such as easy adding and subtracting of time while in progress.

A digital microwave oven controller, by any measure, is an inferior product. The reasons it might be preferred are (1) to reduce component count and save cost for the manufacturer, (2) to appease consumers with an impression of a product in the “advanced” style, neither of which are excellent reasons.

This is a mistake, and the inefficiency introduced via the digital controller is only ameliorated somewhat by some oddities of the device (enter 99 and the microwave runs for 99 seconds, enter x:99 and the microwave runs for 60x+99 seconds, etc.). So between 60 and 99 seconds, you manage to save a keystroke but no more. (Ok, also between 10 min. and 10 min. 29 seconds.)

This reminds me why I dislike digital book readers, too. They suffer from the analogous problem versus paper books with regard to page turning.

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An elementary, nice lemma relating to the optimization of multivariable functions says that the smallest “big thing” is still bigger than the biggest “small thing”, in other words,

\(\min_x \max_y f(x,y) \ge \max_y \min_x f(x,y)\).
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