Tuesday, June 30, 2009
12 or 13? How it works
Check out this little puzzle where the number of people in the picture changes from 12 to 13 depending on the arrangement of the pieces. Where does the extra man come from/go to?
The secret is that the men do not stay the same size. Each is larger when there are only 12 men than they are when there are 13. It is especially noticeable with the man standing at the far left, whose head is neatly chopped off when 13 men are showing. Look carefully at their feet as well. They get noticeable smaller for the 13 man version. The picture is drawn very skillfully so that when the pieces move, each man changes size, by 1/13th of a man, but the black and white sections match in plausible ways.
There are many puzzles like this. I recall one when I was a child that had China men (I know, so un-pc) running around the world and their number changed (I think from 11-12) depending on how an inner circle of the picture was turned. I found the solution by looking closely at their beards and swords, both drawn about the same width, length, and black color. They all got longer when the smaller number of men were showing. Their faces got larger too.
If you are unconvinced, imagine laying out 12 dollar bills, side-by-side, each 1/13th of a bill lower than the previous one, cutting them across the middle, and then shifting the lower half one bill to the left. What you'll get is 13 bills that are 12/13th of a bill in length, with a missing piece at the cut*. The puzzles are just more complicated versions of the same principle.
And personally, I've found this solution fabulous in it's subtlety and simplicity, and a good proof that the solution to a mystery need not diminish the wonder of it.
*Yes, people have tried to counterfeit money this way. I do not recommend it, as it is easy to spot, highly felonious, and the only way the communists really could win.