Stamped On

I’ve had a bit of a rough week.  My job has been stressful, my relationship imploded, and my bracket‘s final four is completely busted.  It was one of those weeks that makes me want to abandon work, romance, and armchair basketball commentary all at once and abscond into the hills, to make a living carving small bear totems out of pine trees felled by lightning.

Those weeks (and, yes, this one included) are generally ended by the realization that WiFi is pretty hard to come by in the wilderness.  And that I really don’t have any woodworking skills beyond third grade summer camp.  But what really kept me going this week — what really allowed me to block out everything else that was going on — was a simple friend, who’s always been there and never really changes that much, no matter what’s going on in my life.  Who’s always right.  Who’s always logical.

I’m talking, of course, about math.  (Har har har, I know.)  Like any good week where I’m feeling stamped on, I found a great math problem at work to distract me.

Without getting into specifics that could, in turn, get me into hot water, I’ll summarize the problem by saying this: I needed to be able to make a lot of differently sized… things.  Let’s call them Lego castles.  But I only could pick from one or two different Lego bricks.  Obviously, if I used bigger bricks I’d be able to build bigger castles faster and more efficiently.  So which (suitably large enough) bricks would let me build the widest range of castles?

Wikipedia calls this the coin problem, but since American numismatics has way too many common factors — and, of course, for the linguistic congruity with my emotions — I prefer to call it the stamp problem.  Of course, I didn’t know this when I started thinking about it, so I had a lot of trouble googling a solution.  “Two integers make lots of bigger integers” doesn’t return very many solid results.

Anyway, the stamp problem goes something like this:

Imagine you have two big piles of stamps.  One pile is made up of stamps worth 3¢ each, while the other pile consists entirely of 5¢ stamps.  This year, postage is 8¢ — but the postal service has decreed that postage will increase by 1¢ each year for the rest of time (thanks, Obama), and you must pay the exact price.  So next year, postage will be 9¢, the year after that 10¢, then 11¢, and so on.  Since you only have 3¢ and 5¢ stamps, what years won’t you be able to send letters because you can’t hit the exact cost of postage?

The answer is never.  You will never be unable to send a letter, because any positive integer greater than 7 can be formed with some combination of 3s and 5s:

8 = 3 + 5
9 = 3 × 3
10 = 5 × 2
11 = 5 + 3 × 2
1,236 = 5 × 246 + 3 × 2
etc.

This is a ridiculously cool phenomena associated with coprime numbers — numbers whose only common divisor is one.  Two coprime numbers a and b can combine to form any positive integer whose value is greater than ab – (a+b).  Combinations of 3 and 5 can make any number greater than 7; 2 and 7 can make any number greater than 5; 73 and 182 can make any number greater than 13,031; and so on.

Mind = blown

There’s a beauty here, lurking behind the guise of ordinary numbers.  If you’ll let me step away from the main point of this post (MATH IS COOL) to the ancillary topics I’ve raised (EMOTIONS ARE TERRIBLE WHY DO I FEEL THEM), I think this is why I like math, and why I find it comforting in some weird, all-too-stereotypical way: it always makes sense.  There’s a reason to it, a structure — even times when I can’t see that structure at first, as was the case of the stamp problem, I know that the underlying architecture is beautiful.  I’m harder pressed to believe that about the other things in my life.

Most of the time, I enjoy the lack of structure.  I wouldn’t want to know what’s about to happen to me, and certainly am not insane (and/or religious) enough to think there’s an overriding reason behind everything that happens to me.  The fact that there’s no underlying architecture means I’m free to build my own structures and connections, free to construct my own edifice.  But when I feel stamped on (promise, last time I’ll use that idiom), when the weight of all the chaos frothing around me gets too heavy to bear, when the people that I care about are suddenly confusing and nauseatingly painful and impossibly distant — well, it’s nice to know that at least something in this universe still makes sense.

Even if that something is just addition.

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