The Volokh Conspiracy

Say that the only point-scoring events in a football game are field goals (3 points) and touchdowns with one-point conversions (7 points). Some point totals cannot be scored in such a game — for instance, 1, 2, and 4. What is the highest integer point total that cannot be scored using just 3-pointers and 7-pointers?

Now say that we exclude field goals, but allow touchdowns with missed-conversions, so the only point-scoring events are 6 points and 7 points. What is the highest point total that cannot be scored using just 6-pointers and 7-pointers?

And now let’s generalize. Say that there are two point-scoring events, one which yields a points and one which yields b points. If a and b have a common divisor, then of course there are an infinite number of positive integer point totals that can’t be scored; for instance, if all you have is 4-pointers and 6-pointers, then all the scores will be even, and any odd score will be unachievable. So let’s assume a and b are relatively prime, which is to say that they don’t have any common divisors. What is the highest point total that cannot be scored using just a-pointers and b-pointers?

UPDATE: Thanks to commenter Nick, I now know this is the Frobenius coin problem.

I was shocked at the number of people who took the view that 0 was neither even nor odd. (I was even more shocked by those who thought 0 was odd, and by those who thought 0 was both even and odd, but there were comparatively few of those.) Under every definition of even that I’ve ever seen, and under every one that to my knowledge has any mathematical utility, an integer is even if it is divisible by 2 with no remainder. 0 is divisible by 2 with no remainder (0/2=0). Therefore, 0 is even. End of story, though if you want much more of the story, read this monster comments thread.

But what shocked me even more was a link to McGraw-Hill’s Catholic High School Entrance Exams p. 213 (2d ed. 2009), which asserts (twice) that “The number zero (0) is an integer but is neither even nor odd.” As I said, this departs from all that I’ve ever seen of actual mathematical definitions. And the material in the book is actually inconsistent with that very definition; for instance, later in the page, it says that

(even integer) +/- (even integer) = even integer
…
(odd integer) +/- (odd integer) = even integer

But that of course is wrong if 0 isn’t even, and right only if 0 is even. (Consider 2-2 and 3-3.) And of course these equations, and many others, are part of the reason that having 0 be even is such a useful definition, one that mathematicians have settled on.

In any case, this assertion in the book can’t be doing its readers any good. I tried to find an e-mail address to which I could complain, but I couldn’t. If any of you can let me know whom I can contact on this, I’d much appreciate it.

I had discussed this before, but I thought it would be good to do a quick survey on this. Please note that this is not a trick question. Also please give what you seriously believe to be the correct answer; and please vote even if you think the answer is obvious.