Here in the Philadelphia area, we have an oddly-named chain of convenience stores named Wawa.
At Wawa, you can buy coffee for the following prices: $1.09, $1.19, $1.29, $1.39 for 12, 16, 20, 24 ounces respectively. This makes sense -- basically you pay $.79 for wandering around in their store taking up space and such, and then 10 cents for each four ounces of coffee.
However, things get weird if you bring your own cup (I'm talking about the "travel mug" sort here, not a paper cup). Then 12, 16, 20, 24 ounces cost $0.85, $0.95, $1.05, or $1.15 -- so far, so good. You save twenty-four cents by bringing your own cup.
32 ounces, in your own cup, is $1.25. So now they're really starting to reward you for buying in bulk -- another ten cents gets you eight more ounces.
But then guess what happens? 64 ounces costs $2.99. That,s right -- I can fill two 32-ounce cups for $2.50, but filling one 64-ounce cup will cost $2.99. If you extrapolate the linear trend from 12, 16, 20, and 24 ounces, 64 ounces should cost $2.15. If I had a sixty-four-ounce travel mug, I'd go in there, fill it up, and try to get it filled for $2.50 just to see how the cashiers explained it.
Perhaps they're trying to say that you really just shouldn't be drinking that much coffee. I'd have to agree -- and I'm a mathematician.
Another argument is that perhaps they are attempting to discourage people from taking that much coffee because then there's less coffee for the people after them, and people won't be happy if the store runs out of coffee. This may be true -- it seems a bit doubtful, though, since a typical Wawa store might have a dozen or so pots of coffee at once, each holding 64 ounces or so.
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I remember shopping for a particular product. Individually, it was priced at $9.95. However, a three-pack cost $29.95. I bought three separately, and kept my dime. That'll show them!
At some time in the early 1990s, the fares on the Bay Area Rapid Transit system did not obey the triangle inequality. There was some trip you could take where you would pay less by going to intermediate stop B, getting out, and then getting back in again, than you would by going direct.
As a mathematician, I was incensed.
That anti-triangle inequality property still holds on the train system in New Jersey.
Newark Airport to New York City's Penn Station is a popular route, and costs $15.00.
Getting a ticket from the Airport to Newark Penn Station (which is on the route between the two) plus a ticket from Newark Penn to New York Penn costs $11.75.
Mark,
I am almost certain that airfares don't obey the triangle inequality.
Then again, airfares change so often and are so complicated that it would actually be more surprising if they did obey the triangle inequality.
Locally, the university book shop have a sale room which has a permanent "buy three and get the cheapest one free" policy. Both in print, and on the radio, the extended consequences of this offer have been explained: "buy six, and the cheapest two are free", "buy nine, and the cheapest three are free".
Interestingly (?) non-mathematicians often don't spot the problem.
On the air fares issue, not only do they not satisfy the triangle inequality but figuring out the best airfare could be undecidable.
And oh yeah, I challenge the identification of the liquid that one can purchase at Wawa as "coffee".
Michael Albert, re "Interestingly (?) non-mathematicians often don't spot the problem" you're right!
I laughed when I read it, but when I repeated the story to my buddy (an electrical engineer), he had to have the punch line explained to him.
um can someone explain the punchline for the sake of non-mathematicians reading this blog? :P
Think about it :)
Hint: If you are buying six books it might be better to take a friend along with you to the shop.
I experienced an extreme version of the train triangle inequality when i had daily commitments in a town 500 km from where I lived.
By buying tickets so that the travel route went from town A to town B that is 400 apart, then from B to C (200 km), and finally C to home (400 more km) I would get home for 50% of the price from A to home directly.
Though it would take 10 hours or so.
Valentine's last comment (vis-a-vis air-travel time) kinda confirmed a suspicion I had all along, viz. that the triangle inequality is not necessarily obeyed if our metric is distance (or volume, say) but that it holds if the metric is time! And this, in fact, explains all the anomalies pointed out thus far. For instance, the time taken to drink 64 ounces of coffee is less than or equal to the time taken to drink coffee from two cups each holding 32 ounces of coffee! The same argument holds for train fares and air-travel. In each case, you pay more money (or the same amount) if you save time! It's weird thinking this way, but it works except perhaps on computer networks: it may take more time for a packet to travel from A to C than it may take for it to travel from A to B and then from B to C!
Oops, sorry! In my previous comment, on the 4th line, I meant to say "... if our metric is fare..."
Regarding the "buy 3n books, get the cheapest n free" thread: it can be very frustrating for someone who confesses that he doesn't get the punchline to be told to run along and "think about it". I can imagine, for instance, that the way the "joke" was framed might make one suspect it was the *bookstore* that somehow stood to lose through the ineptly worded advertisement. Under such assumptions, the "joke" becomes baffling, and the mathematician, who makes invidious comparisons to non-mathematicians and refuses to divulge the punchline, becomes persona non grata.
Suppose for example that the customer is interested in buying six books priced at 7, 6, 5, 4, 3, and 2 dollars. It then makes more sense to buy the 7, 6, 5 in the morning and the 4, 3, 2 in the afternoon (saving 5+2 dollars), than to buy all six at once (saving 3+2 dollars).
Invidious? Hmm, more like wryly amused. It's a very rare occurrence indeed when the mathematician's approach to a "real life" situation is more effective than that of "the man on the Clapham omnibus". Sorry if my choice of words led to an unshakable conviction concerning my hidden intent.
As for providing the answer; had the commenter provided a contact address I'd have been delighted to. As it was, I judged that there might be others who would appreciate thinking about the situation without an overt spoiler. I did feel that my hint was pretty broad, and in my personal experience such a hint is often more appreciated than a direct answer would be. I think a fellow called Socrates might once have had something to say about that.
Unshakable conviction? Nah, I meant it more in the sense of the expression "all comparisons are invidious" -- I just think referring to non-mathematicians as a class is treading on dangerous ground, and is bound to invite misunderstanding. Thanks for the explanation, and sorry that what I said came off as ad hominem.
But wolz *did* ask for an explanation, and you refused. You may *think* your Socratic hint was both sufficient and appreciated, but that's only if they got it; if not, they may wind up feeling even more stupid and frustrated than before. It takes guts to admit that one (maybe speaking for many) isn't getting something, and we mathematicians/teachers need to respect that.
The reason for this discrepancy in coffee price scaling is that people are bad at math (they really don't target mathematicians with this stuff - there are too few of you to really make a difference in their bottom line). My guess is the 64 ounce size is so large it looks like a bargain at anything around $3. No one even thinks about doubling the 32 ounce price - too hard to do in their head (and most people wouldn't even think of trying it). The relationship between perceived value and size of a coffee mug gets distorted really fast. Give someone a huge mug that is 4 times the size of the 64 ounce one, and they will tell you it's probably worth $20 (or more).
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