Self-Explanatory

A counterexample, in Mathematics, is a mathematical object that, merely by existing, shows that a proposed theorem can not be valid, the failure of validity often being quite strikingly counterintuitive. The construction of these objects serves an intellectual purpose, in that they set limits on that which is provable. They serve an aesthetic purpose, in that they can be strikingly beautiful in their own way. They also serve an emotional purpose, in that they can be wonderfully effective in quieting whiny students, when the students confront the grader over the points they lost in making an invalid assumptions in some of the proofs they submitted, and say "but thaaaaat's obviouuuuuuus!", reducing the probability of the hapless grader developing an ulcer by the end of the week. Of course, it's a dead on certainty that he'll have one by the end of the year, but by then he should be able to go home and have his relatives nurse him back to health, just in time for another year of pulse pounding action in the back of the library, red pen in hand and papers stacked high. Yes, live the dream. Oh, dear G-d.

This is a companion to "Doing Your Homework", a blog on which I'll be working a variety of math problems.