April 01, 2005

Get Rich Quick!

People occasionally ask me why, if I spent all that time studying math in college, I don't put my skills to good use and actually figure out ways to get filthy, stinking, uproariously rich. So, during my lunch break today, I came up with a foolproof scheme to get a whopping 66 percent return on the dollar. No joke! Details and nifty graphical display (thanks to the magic of Microsoft Paint) below the fold...

The basic scheme involves turning $3 into $5. How? Well, first we need to believe the (totally unverified) rumor that stores will accept a dollar bill if over 51 percent of it is intact and topologically continuous. In fact, let's assume that you need a little bit more than 51 percent—say, 60 percent—to pass muster. So you lay out your three dollar bills on the table, and cut them up along the black lines showed in figure 1 below:



Now, take all those green triangles you cut out of the dollar bill up on top, and with a little bit of scotch tape, attach them on to the other pieces as shown in figure 2:



Et voila! You now have 5 viable dollar bills, each one continuous and with 60 percent the area of a full bill. Obviously you'd need to get the geometry just right so that the triangles all fit and each have an area equal to 10 percent of a full bill, but that's not too hard to finagle. The flipside here is that you probably couldn't do this with $20 or $100 bills, since cashiers would likely give those bills a bit more scrutiny and be loath to accept something that looked like Dracula fangs or an "X". But when the stakes are low—and the stakes are low with a $1 bill—the local convenience stores should accept your odd shapes so long as you appear supremely confident when you slap them down. At any rate, clearly this isn't actually a scheme to get filthy rich, but it should lower the price of my beloved 7/11 Hot Pockets dramatically...

UPDATE: Just to address an objection raised in the comments, the new bills seem perfectly continuous to me. Topologically speaking, all I'm doing is mapping from one space to another by creating a bunch of equivalence classes. All surjective identification maps—which is what this boils down to—are continuous. In fact, my newly minted set of $5 even has its own unique quotient topology, which is cool though largely useless in this context. And yes, granted, this argument might not fly with the storeowner across the street when I hand him a bunch of jagged dollar bills swaddled in tape, but it would if he were perfectly rational...

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-- Brad Plumer 3:48 AM || ||