The Twelve Man / Thirteen Man Problem

If you enjoy Sam Loyd, you may also want to check out my post Sam Loyd's Trick Mules. - MB

Every few years the "twelve man / thirteen man" puzzle makes its way around the Internet. And every time I see it I am baffled.

If you don't know what I'm talking about, click here. That's an animated gif, so keep watching until things move. When the image first appears, count how many men there are. Then, after the top halves swap, count them again. The first time you should count twelve; the second, thirteen.

I've long suspected that I could figure out the trick if I really applied myself but, slacker that I am, consistently given up after a minute or so.

Well, I came across the "twelve man / thirteen man" illusion yet again today. But this time there was an accompanying image by Matthew Sturges, one that colors the men and shows both their start and end positions. I took his image, added numbers, and finally think I can see what's going on here.

There's two reasons this is so hard to wrap your mind around, I've concluded. The first is that the drawings look unrefined, which both disguises the fact that the solution is very subtle, and gives the viewer few key features to use as reference. About the only clearly identifiable body parts are heads, torsos, arms, legs, crotches, and feet. Note that their hands are all hidden behind their backs -- crafty, that.

The second reason this illusion tends to defy analysis, I think, is 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. Pick a man in the first picture, look to see where your chosen body part is, and then look to see where it ends up in the end configuration. Now repeat this for all twelve of the men. 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: #1 -> both #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 thirteen different composites. Thus, you can't point to any one person in the second images and say "he's the new one."

[There used to be a few more paragraphs here describing which men in the first picture contributed what to whom in the second, but Jon's illustration, in the update below, neatly summarizes everything.]

If you're still not getting it, take a look at this simplified version of the illusion, where I magically turn five lines into six:

The "twelve man / thirteen man problem" operates on exactly the same principle, although it's cleverly convoluted to make it seem like there's more going on. Notice, for instance, that, on the average, the men in the second picture are shorter than the men in the first, as is the case with the lines above.

Incidentally, this is a variation on Sam Loyd's famous "Get Off The Earth" puzzle, which you can read more about here.

Update: Good gravy, I can't believe I'm got to spill yet more virtual ink on this. But I did say I wanted this to be the definitive page on the subject, so here we go.

Some folks in the comments and claiming that the 12-13 Man Problem is waaaaay more straightforward than I am making it out to be. "Look," they say, "you have 12 men in the first picture. You split them into 24 halves and recombine 22 of those halves into 11 people. Then -- and this is the entire trick -- you point to the remaining two halves and claim they are full people. 11 + 2 = 13 men. In the final configuration, the two 'half men' are #1 and #13, each of which gives up a half and doesn't get one back."

They people making this argument are absolutely right: that's how the trick works in principle, and I said as much in giving the illustration of lines. But they are ignoring the key element that makes the 12-13 Man Problem different from the line example. If you bisect a line you can truthfully call each of the resultant halves a "line," but if you cut a person in half you can't claim that you haven't really done anything because each of the two halves is a person itself. (Believe me, when I used this line the police were not impressed ...)

The 12-13 Man Problem is so baffling because each of the final thirteen men looks like a full person, even the two "half-men." And it's not just #1 and #13 that are involved: if you were to take the missing half of #1 and the missing half of #13 and put them together, one of your men in the final configuration would consist of nothing more than a scalp on a pair of feet.

No, all the men are altered. And luckily for me, Jon over at Corporate Superhero has created an image that shows how:

In his words: "Basically, the puzzle works by cutting each person in two, taking a small slice of them (1/12 of their height) and passes it over to the right until after 12 people you end up with a whole extra person. Then the creator mixed up the order of the people so that you couldn't see what he did."

Thank you, Jon -- your picture is worth several thousand of my words.