# 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?