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I got this puzzle from YouTube. I would watch the video first, but I'll include screen captures and a description below so we can discuss it.

13 Man Arrangement

In the beginning of the video he starts with a picture consisting of 13 cartoon men.

13 man arrangement

After counting them (accurately) he then rearranges them using these cuts:

13 man arrangement with cuts illustrated

12 Man Arrangement

Here is a good image of the cuts right before full placement:

12 man arrangement with cuts shown

Full placement:

12 man arrangement

If you got here without watching the video, the curious part is that
there are now only 12 men
.

WHY DID WE LOSE ONE MAN??


My Analysis

I realized I needed to narrow in. I chose shirts and pants to focus on. I originally tried just shirts or just pants, but because of where the split is, I wasn't getting anywhere. Choosing shirts and pants allows me to split the men in 2 like the cuts do.

The way I see it, logically, if the number of pants and shirts don't change, then neither should the number of men.

But if you count them in the separate images, you see that they do!

I thought if I wrote it down and played with it I might see what I am missing. I went from left to right, bottom to top (pants then shirts) representing the colors with "b" and "w" ("black" and "white" respectively).

NOTE: Because of the placement of the men, it made more sense to go bottom to top if you try splitting them. If you tilt your head to the right you can see it matches the image.

  • 13 Man

    • bw/ww/bw
    • wb/bw/wb
    • bw/bw
    • ww/bw/wb
    • bw/bw
  • 12 Man

    • bw/ww/wb
    • wb/bw
    • bw/bw
    • ww/bw/wb
    • bw/bw

This is where I'm stuck!!

Side Note:

This reminds me of another puzzle (I can't seem to find it now) where an image of a comb has a number of teeth (well say 13), and the way the image of the cut (much like this one) if you rearrange it you end up with 12 (or something) by increasing the length of the teeth.

I think a similar principle applies here, but I just can't see how.

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  • $\begingroup$ Try looking at the bottoms of the shoes and the tops of the heads. $\endgroup$ Dec 28, 2023 at 19:40
  • $\begingroup$ This has been well explained already. A good explanation is here: youtube.com/watch?v=rd5hr70iWgM $\endgroup$
    – Hilmar
    Dec 29, 2023 at 15:45
  • $\begingroup$ That's a good puzzle. Has anyone got a link to a decent quality version of the original image with cut lines? $\endgroup$
    – Transistor
    Dec 29, 2023 at 16:12

3 Answers 3

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Here is a pictorial way to see what is going on. In its simplest version, it works like the following picture:

enter image description here There are 12 tall rectangles, each 13 segments high. By shifting the top half of the picture one step to the right you form 13 rectangles, each 12 segments high.

Having these rectangles in order like this makes it rather obvious, so they instead rearrange them to be spaced five apart, so that you have to (cyclically) shift the top half five steps to the left instead of just one step to the right.

Here are the heights of the parts in the 12-man arrangement:

    1  6  11 3  8 |0  5  10 2  7  12 4  9
    12 7  2  10 5  0  8  3  11 6  1  9  4

Note that each man has total length 13 because every column adds to 13 (except for the empty spot near the middle). By swapping the two pieces, the top halves of the men are shifted so that each is replaced by the next smaller one, and the largest part becomes a whole man on its own. So it becomes this arrangement, where there are 13 men each of height 12:

    0  5  10 2  7  12 4  9| 1  6  11 3  8 
    12 7  2  10 5  0  8  3  11 6  1  9  4

It is not possible to point to any particular man that has (dis)appeared, because each man consists of two parts which end up in two different men when the pieces are rearranged. The 12 men are disassembled into two parts and those parts rearranged to form 13 different men.

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  • $\begingroup$ Ah, yes! Is this is great! You just barely beat me to my own breakdown. It was bugging me too much. This is a great illustration! $\endgroup$
    – Hawkeye
    Dec 28, 2023 at 9:17
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    $\begingroup$ Animated version where you can watch all the height differences happen together. $\endgroup$ Dec 28, 2023 at 13:46
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    $\begingroup$ "It is not possible to point to any particular man that has (dis)appeared", but it is possible to point to the two men who acquired a bit more feet and bit more head respectively $\endgroup$ Dec 29, 2023 at 6:33
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The easiest way to notice what's going on is to look at the heads and feet of the people. In the first arrangement, the person on the far left has a perfectly flat head. In the second arrangement, his head is round because a small slice of somebody else's head has been moved on top of it. Likewise, the sixth person from the left has a perfectly flat pair of shoes before moving the pieces, and that same person has a thicker, rounder pair of shoes after the move. Each person is becoming slightly taller, by swapping its top half with a larger top half from a different person.

Of the 13 initial people, 11 of them are made of two "half-people", but the remaining two (the ones mentioned above) are only a single "half-person" each, for a total of 24 "half-people". After rearranging the 24 "half-people" such that each half is paired with another half, only 12 people can be seen.

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  • $\begingroup$ While this is what I was sort of eluding to, I don't see it. You can explain it this way and it makes logical sense, but it isn't making pictoral sense to me. I can't help but think we're missing something. $\endgroup$
    – Hawkeye
    Dec 28, 2023 at 3:18
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    $\begingroup$ What are you confused about? $\endgroup$ Dec 28, 2023 at 4:08
  • $\begingroup$ I guess I couldn't visualize it until I spent more time (I'm a visual guy) and I think I needed a break down of the steps. Even @JaapScherphuis answer (having a great illustration, being similar to what I mentioned about the comb trick) didn't satisfy where the body parts were coming from. So I hope you're not offended that I included an answer following my efforts. I'm finally satisfied that I understand it now. Lol. $\endgroup$
    – Hawkeye
    Dec 28, 2023 at 9:33
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While I believe @codewarrior()'s answer is correct, it just wasn't intuitive to me.

I felt I needed a breakdown of exactly what was happening.

UPDATE: @JaapScherphuis' answer has a great picture that helps too!

Similar to the comb trick I mentioned,

But I needed an explanation as to where spare body parts are coming from, and it turns out there are a few tricks used

that are a little more crafty than simply borrowing a peice.

The following is my effort to break it down into steps

I first found it more intuitive to do it

in reverse order of the video because I now beleive the original image has 12 men.

So I numbered the men in the picture:

12 men numbered

And from this image I will refer to them by their numbers in the list below.

Step 1. Empty space lands on 1's head
(freeing up the crown of a head)

Step 2. 1's crown lands on 8's eyes
(freeing up a crown and forhead)

Step 3. 8's eyes land 4's upper nose
(freeing up a crown to a nose)

They do a neat trick here:

Step 4. 4 uses his upper nose landing on 11's shoulders to convert the neck of 11's shirt into 4's new jaw
(11's whole head has been freed up)

Step 5. 11's head lands halfway down 6's chest
(freeing up a head and shoulders)

Step 6. 6's head and shoulders land right above 2's belly
(freeing up a head and chest)

Step 7. 2's head and chest land halfway down 9's belly
(freeing up a head and what will work for a torso)

Step 8. 9's head and torso land halfway down 5's thighs
(freeing up what will work for a body above the knees)

Step 9. 5's head to thighs land on 12's knees
(freeing up a head to what will work for a body down to the ankles)

Step 10. 12's head to ankles land on 7's feet
(freeing up a head to ankles with longer legs)

Step 11. 7's head to ankles land on the soles of 3's feet serving as thin shoes
(freeing up a whole man with thin shoes)

Step 12. 3 with his thin shoes lands on 10's shoes making them thicker shoes
(freeing up 10 as a whole man with thin shoes of his own)

Step 13. 10 lands on empty space
(a new man)

NOTE: the part I couldn't wrap my head around before was that

A number of these steps used visual illusions and repurposed parts to serve the new smaller bodies. That's the part that was bothering me because it wasn't as simple as just shortening a line; they used tricks to free up body parts.

I don't usually add an answer on top of another's regarding a question that I asked, and I must give credit to @codewarrior()'s answer for confirming my suspicions

as I alluded to about the comb puzzle mentioned in my question.

But I thought I would share what I eventually came to (that finally satisfied my confusion that has kept me up so late.)

P.S. I welcome more answers to perhaps even better explain this facinating puzzle.

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