My wife is out of town this weekend, so I decided to stay up late Saturday night and scratch a figurative itch that’s been in the back of my mind for a few years — XKCD’s 3-velociraptor problem. (I spent early Saturday night at an Ira Glass show at Wolf Trap, so apparently I don’t turn into a complete hermit when the mate is away.)
I’m not a mathematician (my brother got those genes), so I reached for my trusty brute-force, computer-simulation experience to solve the problem. At first glance, you’d think that you want to run toward the limping velociraptor at the top. And you’d be right, but to a surprisingly small extent. The optimal angle is about 32 degrees–only two degrees different from the angle you’d run to bisect the vicious predators if all velociraptors were healthy. Of course, there’s a symmetrical solution on the left side (about 148 degrees) as well, as demonstrated by the gif below. You live for all of 3.06 seconds, and get about 18 meters away.
The reason that the answer differs so little in the top-VR-is-injured variation is that over three seconds, the two healthy beasts only reach 12 m/s, which is only 2 m/s faster than the injured animal. I was disappointed when I realized that Randall probably didn’t solve the problem before writing the comic–if he had, I would have expected a more interesting conclusion.
Thus, in the vein of XKCD’s “What If” posts, let’s change some of the parameters. First, obviously humans can run different directions whenever we want–that’s the benefit of being the world’s (current, supposed) smartest species. However, because the aforementioned injury is barely holding the top raptor back and because you’re still much slower than it (6 m/s vs 10 m/s), the solution in this variant doesn’t change.
But what if you could out pace the lame duck (so to speak)? Setting the maximum speed of the top therapod at 5 m/s changes things dramatically. You should run at the top predator at only a slight angle (~10° from true north), brush right past it and head for the hills. While the lower raptors will catch you (in 3.95 seconds to be exact), in my gif our terrified human runs right off the 30mx30m screen to arbitrarily-defined safety!
Now we just need the Costa Ricans to bomb the island.