The missing man and the missing square – the answer

There are a few puzzles hiding away in the depths of this blog that I havn’t ever got round to providing the answers for. Today I will provide the answer to two of my favourite puzzles, the missing man and the missing square. Firstly, here are the puzzles again

In the missing man puzzle, count how many men there are in the animated image. There should be 12. When the top halves of the men switch, count again. There should now be 13.  Where does the extra man come from and where does he go?

In the missing square puzzle, when a series of shapes is rearranged, 1 square is missing. How is this possible?

The answers are below:

Answers: Both of these puzzles do something very similar, they seemingly make something disappear or reappear. As we shall see, they both achieve this by fooling us into thinking we are seeing something we are not.

In the missing man, we think we are seeing 13 men become 12 men. We are not. Or more precisely, we are not seeing 13 men become 12 men which are identical to the original 13. You see, when the 13 men turn into 12 men, each of the men gains something… literally. Specifically, each of the 12 men gains part of the man who goes missing!… 1/12 of the man who goes missing. For example, the man on the left gets the missing man’s hair. By distributing a small part of the missing man equally between all the 12 remaining men we don’t notice any difference. The fact is that our definition of a man in this puzzle is flexible enough (the rough drawings help with this) to be able to ignore these changes and make us think we are seeing the same thing before and after the man goes missing.

In the missing square, the exact same is happening. We think we are seeing a triangle become a triangle (with a missing square). Again, we are not. The diagonal side of the “triangle” is not straight (it is not really a triangle). Instead it has a bend in the middle. In the first image, this bend goes inwards. in the second image, this bend goes outwards. What this means is that there is a slight difference in area between the two shapes. This difference in area corresponds to the area occupied by the missing square.

So to summarise, while we think we are seeing 13 men turn into 12 men and a triangle turn into a triangle (with a missing square), we are not. We see 13 men become 12 slightly taller men and a “triangle” with a side bulging outwards become a “triangle” with a side bulging inwards. The fault lies not with our eyes, but with our definition of the things we see.


8 thoughts on “The missing man and the missing square – the answer

  1. Sorry Michael, some of the men do get shorter, but the feet is the problem, 25 feet one way 23 the other way(one foot always behind the bottom row right) have worked for 2 wks, made pictures cutouts enlargements cut my paperwork like the computor. Always the same 25 feet and 23 feet, I hope one day it can be solved….Tks, Johnny

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