<< Good Luck With That |

Every few years the "Twelve man / Thirteen man" animated gif seems to make its way around the Internet, and each time I see it I am baffled. If you don't know what I'm talking about, click here.

I've always suspected that I could figure out the trick behind this puzzle if I really applied myself, but after a few seconds of trying to noodle things out while watching that image move all around, I've always given up.

Well, it's going around again. But this time, Matthew Sturges created an accompanying image, one that colorizes the men and shows both their start and end positions. And with this, I think I can finally see what's going on.

I took Sturges' image and added numbers. Here it is:

There's two reasons this is so hard to wrap your mind around, I've concluded. The first is that the drawings look unrefined. This disguises the fact that the solution is very subtle, and serves yet another purpose. By making the men crude cartoons, we are given few key features that we can use for reference: about the only clearly identifiable body parts are heads, torsos, arms, legs, crotches (i.e., the point where the legs meet the torso), and feet. (Note that their hands are all hidden behind their backs -- crafty, that.) The second reason this illusion tends to defy analysis that because there is no "smoking gun" solution to it, something you can point to and say "Aha! Here's where the 13th man comes from." That's because the thirteenth man comes from all twelve of the others.

Look at the start configuration, and note that there are twelve of each body part: twelve heads, twelve torsos, twelves pairs of legs, etc. Now look at the end configuration, and note that there are thirteen of each body part. That makes it seem as if a thirteenth person has somehow materialized.

But now narrow your focus. Instead of looking at the whole pictures, just pick a single body part, and look to see where it is in the start configuration, and where it ends up in the end configuration. In all cases -- and this is the key point, kids -- one of the twelve instances of a body part in the first picture is bisected and used twice in the second.

For example, let's look at faces. Man #1's face in the first picture is below the divider, so it remains with man #1 in the second picture; man #2's face along with the rest of his head) goes to man #9; man #3's face goes to man #10. So far so good. Now look at man #4. His face is split in half, with the top half going to man #11, and the bottom remaining with man #4. In other words, the single face owned by man #4 in the start configuration is now two faces in the end configuration; in other other words, where there were twelve faces there are now thirteen.

Pick another body part, do it again -- and again you'll see that one of the body parts in the first picture is split and used as two in the second. Here's the breakdown:

  • Hair: both #1 -> #1 & #8
  • Face: #4 -> both #4 & #11
  • Arms: #2 -> both #2 & #9
  • Torso: #9 -> both #5 & #9
  • Crotch (i.e., point where legs meet torso): #5 -> both #5 & #12
  • Legs: #12 -> both #7 & #12
  • Feet: #10 -> both #6 & #13

So in the second picture we get a new head of hair, a new face, a new pair of arms, a new torso, a new crotch, a new pair of legs, and a new pair of feet -- all of which adds up to an entire new person. But these parts are distributed amongst seven different composites. Thus, you can't point to any one person in the second images and say "he's the new one."

Now let's look at who get's what. We'll start with man #1. Man #1 splits his hair into two, keeping the lower half and contributing the upper half to man #8. And imagine that, in order to put #1's scalp onto #8's head, we first had to remove the top of #8 head that he started with. At this point both man #1 and man #8 are complete (man #1 doesn't get a top in the second configuration), so we have two whole persons -- and we're still holding the top of #8's head, ready to move it somewhere else. Just like that we've gone from two men in the first picture to 2 1/6 men in the second.

Now let's move man #8's hair over to man #4. And we'll move the top half of #4's head over to man #12. As we noted before, man #4's face serves double duty in the final configuration, so now, after only four moves, we are alreasy "up" both a head of hair and a face. Can you start to see how we're going to "build" the thirteenth man?

Man #11's head goes to man #6. Man #6's head and shoulder's go to man #2. Man #2's arms are split and become the arms for both the final #2 and the final #9. Half of our thirteenth man has been constructed. By the time we get to our last man, #10, we have everything we need except for feet; #10 splits his feet in tw0, becomes man #13, and voila: where there were #12 men there are now thirteen.

Incidentally, this is a variation of Sam Loyd's famous "get Off The earth" puzzle, which you can read more about here and find a pdf version of there.

Posted on April 18, 2005