I don’t have a car. (Don’t need one, don’t want one, don’t want to pay for one, and for a maniac a couple of miles of Finnish winter on a bicycle every day are nothing. As are traffic signs and motorists, too.)
I’m not interested in mechanical things; not in any practical, applied pursuit, at all. (Whether this is because I’m in mathematics, or I’m in mathematics because of this, I don’t know. It’s the mathie/klutz-ditz corollary of the chicken-egg problem.)
Thus I can’t explain why this podcast/radio show called Car Talk is so very fascinating and fun to listen to; but it is. Even if the eventual diagnoses are to my hearing very much like “you should check the braaaaap for water and the braaaaap belt braaaaap brake braaaaap braaaaap uptight wingnuts.”
(Then again, I’ve listened to Linux Outlaws now and then despite never having used Linux for more than four or five minutes, and I’d most probably give a podcast about knitting a listen too if I happened across one. Curiosity is a harsh mistress.)
Now, a recent episode of Car Talk had a puzzle essentially like this: A guy has to remember a phone number. Only the last four digits are different from his own, so they’re the only part that’s a chore. And even they seem easy as they happen to be his son’s age, his daughter’s age, the number he’s behind on payments for some particular purchase, and the number of times his wife’s banished him to a couch.
In addition to this (and ah, the sweet mathematics) he remembers the four-digit number thus formed is divisible by three.
After two weeks he needs the number, slaps the digits together, checks it’s divisible by three, and rings.
Wrong number. And the question to this was, what went wrong?
The given answer was that (as can easily be shown) if a number is divisible by three, it’s divisible by three no matter how you change the ordering of its digits. Thus our guy transposed a number or two, and his divisibility failsafe failed to save him. Plus he now has 23 more combinations to try. (Insert “gasping, hacking laughter“.)
I, in my fevered mathematicaliousness and inability to deal with applications got that far (and that was enough) and then veered into unnecessary speculation, sure that the problem couldn’t have been that simple.
Thus I made it, in my mind, into a non-simple one.
Namely, thought the first, maybe he remembered the order of the numbers correctly.
Thought the second, if so, maybe he failed to account for the numbers changing in the two weeks between the invention of the mnemonic and its application.
Thought the third, of the four digits, three only increase or stay the same (kids’ ages, couch stays). The fourth (payments he’s behind on his purchase) either increases, stays or decreases. Three of the digits (the ages, the payments) can only increase by one each during the available time (one birthday a year; one payment monthly or bi-weekly), though the third of these (the payments) can decrease by one or more. Assuming a non-disastrous and non-come-to-riches person, the two most variable digits (couches, payments) are still most likely to change by single units.
Thought the fourth, a number is divisible by three if and only if the sum of its digits is divisible by three. (From which the digit-shuffling result immediately follows.) Thus the increase/decrease in the digits had to be a total of three units or a multiplicity (0, 3, 6, 9, etc.) in either direction.
Thought the fifth, a total increase of three could mean two birthdays and one couch night. This is not very likely; surely the chump would remember celebrating two birthdays! Come to think of it, he should remember celebrating a single birthday, too; only two weeks had passed. (Remember the kids are of single-digit age; they won’t be holed up in the Catskills with booze and loose members of the opposite/same sex, saying meh to the thought of their parents attending the festivities. Or if they are, they’re in a set of zero measure and we shall not concern ourselves with them.)
Thought the sixth, if we assume no birthdays were had, those two digits are fixed. Now then we have three choices: either our guy spent three, six or nine nights on a couch in the interim — after nine the you leave the one-digit territory, which is not allowed — or he made three, six or nine extra payments. (In both of the nine cases the initial digit was “zero!” which would be kind of a dick mnemonic to use.) As even the threes seem like something one would remember, let us forget these alternatives and take up the last one: couch nights and extra payments canceling each other out.
Thought the seventh, namely, let us suppose our guy spent one night on a couch during the two weeks (plus one to that digit), and made one extra payment (minus one to that digit). That seems like something that is humanly forgettable, preserves the sum of the digits and thus the divisibility by three, and hopelessly perverts the original phone number. (Two nights and two payments or any choice of n nights and n payments works as well, but those creep to the vaguely and conveniently defined “he’d remember!” territory.)
Thus, assuming normal mental processes and a non-screwed-up and non-come-to-riches life for our Guy Average, it seems most likely his number failed because he’d failed to remember one couch night and one extra payment (thus one less payment behind) had occurred during the two weeks after his invention of the mnemonic, and they had influenced the number in a way that did not affect its divisibility by three.
But no, they weren’t after anything as elaborate as that.
And as I think I said before, mathematical types have trouble with applied problems, because sometimes it’s so difficult to know how deep you’re supposed to go. (Can imagine the same of psychologists. “Well he has obviously repressed the memory of his couch episodes because he does not want to make the phone call. I recommend Benzedrine and trepanation.”)