02 | August | 2008 | Continuous Everywhere but Differentiable Nowhere

Tomorrow is my birthday!

I will be turning 27, which is truly the most significant birthday I’ve had to date. This is how I figure it. I was born on August 3rd, and I didn’t get to enjoy my golden birthday, because I was too young to appreciate it. (For those of you not in the know, golden birthdays are when you turn the age of your birthdate, in my case 3).

But shouldn’t turning 27 be even more precious than my golden birthday, because it’s ( $goldenbd^{goldenbd}=3^3$ )? I have decided it is so, and will designate it my platinum birthday.

In totally unrelated news, MIT’s magazine Technology Review has two really good puzzles in their puzzle corner. I think I solved them, but since they are the kind where you send in your answers, I won’t post the solutions here. Just the problems, for you to mull over.

Jerry Grossman has equipped $n$ children with loaded water pistols and has them standing in an open field with no three of them in a straight line, such that the distances between pairs of them are distinct. At a given signal, each child shoots the closest other child with water. Show that if $n$ is any even number, then it is possible (but not necessarily the case) that every child gets wet. Show that if $n$ is odd, then necessarily at least one child stays dry.
Each of logicians A, B, and C wears a hat with a positive integer on it. The number on one hat is the sum of the numbers on the other two. The logicians take turns making statements, as follows:

A: “I don’t know my number.”

B: “My number is 15.”

What numbers are on the hats of A and C?