Average can be meaningful even where total isn’t. Feynman himself gives the complaint I gave.
(Alright, pedantically, there is a sense in which “total temperature” can be meaningful, but it’s subtle: You can meaningfully speak of the total temperature of a fixed number of objects [amounting to just the appropriately scaled version of their average temperature], but you cannot meaningfully speak of the total temperature of an arbitrary number of objects. It is not meaningful to ask whether the total temperature of objects A, B, and C is greater than the total temperature of objects D and E, for example. But still, Feynman’s basic complaint is that “total temperature” is not a simple, useful notion, as the textbook implies; the textbook is just dressing up an ordinary addition problem in a stupidly contrived way.)
You can, if you like, speak of the total temperature of a fixed number of objects [amounting to just a roundabout way of describing their average temperature], but you mustn’t pretend such total temperatures are themselves temperatures (you cannot meaningfully ask whether the total temperature of A and B is greater than the temperature of C, for example (except in the sense of identifying “temperature” with “temperature difference from absolute zero”)), and more generally, you cannot compare total temperatures of differing numbers of objects. There’s a notion of temperature, and there’s a notion of sums of two temperatures, and there’s a notion of sums of three temperatures, and so on, and these are all utterly different kinds of data, as such, until you turn them from sums into averages.
Averages can be meaningful even where totals are not quite; this is the distinction between a convex space (such as temperatures naturally and mundanely comprise) and a vector space (which temperatures do not naturally comprise, unless one takes absolute zero as zero).
This puzzle is used as a joke to suggest John von Neumann’s skill at calculation. After giving the answer immediately, the questioner said “You’re good. Most people sum the infinite series, rather than thinking of the shortcut.”
Von Neumann responded: “What shortcut?”
(Of course it’s a silly joke, since von Neumann was one of the greatest geniuses of this century and would have seen the shortcut at once … or, even on a slow day, immediately upon noting the infinite sum.)
I don’t know if this exactly counts as a math problem, its more like a riddle with math in it, but its my favorite
3 men rent a hotel room for $30 a night. They decide to split up the cost so each man pays $10. Later, the night manager realizes he overcharged them by $5, so he sends the bellboy up to their room with 5 $1 bills. The bellboy decides to keep $2 for himself and refund each man $1. Since each man then paid $9 for a total of $27, and the bellboy keeps $2, that adds up to $29. Where is the missing dollar?
I know that’s the intended answer to the joke, but I’m not convinced that’s the only valid answer. Why doesn’t the two-dimensional quantity of dirt forming the inside area of the hole count as being “in” the hole?
That is the problem. A lot of people would wind up with 25, but the correct answer is 70. The tricky part is, you’re not dividing 30 *in *half, you’re dividing it by one half.