impossibilities in mathematics
Alice laughed: "There's no use trying," she said; "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."
– Lewis Carroll, Alice in Wonderland
Mathematicians are used to believing things that most people would consider impossible or, at least, too outrageous to contemplate, such as the Banach-Tarski paradox. However, there are some genuinely impossibilities, even in mathematics, including trisecting an angle, doubling a cube, and squaring a circle using only a straightedge and compass; finding the center of a given circle with a straightedge alone; deriving Euclid's parallel postulate from the other four; and representing the square root of 2 as a rational fraction a/b. Less well known is this little gem from Gustave Flaubert (1821–1880), who sounds as if he had seen too much of this type of problem in school:
Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?