Yesterday, whilst googling* for tutorials on making Maplets, I ran across a paradox I hadn’t seen since graduate school. It’s a wonderful little illusion, and it reminded me of several similar versions I’ve seen over the years, which I commonly refer to as “something for nothing” paradoxes. So I figured I share four of my favorite such paradoxes with you. Can you figure them out?**
* And yes, “googling” is actually a well-defined, transitive verb, according to the good folks at Merriam-Webster:
** I shall give resolutions to these paradoxes sometime in the future, but if you figure any of them out, please feel free to add it to the comments!
So, without further ado, the four paradoxes.
Paradox the first: the missing square
In which a right triangle is dissected and reassembled to form the same triangle, although now the triangle is missing exactly 1 square unit!
Paradox the second: the extra square
In which a perfect square is dissected and reassembled to form a rectangle. However, whereas the original square had 64 square units of area, the new rectangle has 65 square units of area. Apparently, this is where the missing square from the first paradox went.
Paradox the third: the extra person
Getting something for nothing is a paradox in and of itself, but whereas the previous paradox conjured a mere extra square of area from nothingness, the following paradox manufactures an entire new person from the void! Specifically, the collection below of 12 people will be dissected and reassembled into a collection of 13 people. To witness this transmogrification, click on the image below and watch the subsequent animation.
Paradox the fourth: the missing dollar
In an attempt to restore the status quo, the final paradox will once again involve something missing. This paradox requires no illustration, only the following story.
Three mathematicians went to a convention. They needed a room, but all the hotels were full. They finally found a motel that had a vacancy. They told the Innkeeper they needed rooms. The Innkeeper said “I’ve only got one room left.”
The three mathematicians said “We’ll take it.”
“That’ll be $30.00.”
The mathematicians each pulled out ten $1 bills; they handed the collected $30 to the Innkeeper and went to their room.
After a while, the Innkeeper thought to himself “I’ve overcharged those three men. I should give them a discount for having to share one room.” He called the bellboy over and told him: “Take this money to room 303 and tell the three men there I’m giving them a discount for having to share a room.” He handed the bellboy five $1 bills.
The bellboy took off to the three men’s room. On the way, he thought, How are three men going to split $5? I can help them out by giving them just three dollars. So, in the spirit of altruism (obviously) the bellboy quietly pocketed two of the $1 bills. When he got to the room, he rang the bell and when one of the mathematicians answered, he said “The Innkeeper said to tell you he is sorry for the inconvenience, and offers this refund for your hardship.”
He then handed the man three $1 bills and left. The mathematician gave a dollar to each of his companions, and the three went to sleep.
Since each of the mathematicians received $1 back from the bellboy, each man paid only $9 apiece for the room. That is, they paid only $27 for the room. The bellboy has $2 in his pocket. This accounts for $29 of the original $30 paid. What happened to the missing dollar?