Epimenides, a Cretan, says that all Cretans are liars. This seems to create a paradox: If his statement is true, then it’s false, and vice versa.

The idea recurs in Paul’s epistle to Titus: “One of themselves, even a prophet of their own, said, The Cretians are alway liars, evil beasts, slow bellies. This witness is true.”

But Aristotle, being Aristotle, sees right through the difficulty: “There is no impossibility in supposing that the man habitually lies, but that in this particular instance (in the proclamation of his own mendacity) he is telling the truth.”

Suppose that a person take an even number of coins or counters, or any such in one hand, and an odd number in the other, there is a simple method by which to tell in which hand the even number is. Ask the person to multiply the number in the right hand by an odd number, and the number in the left hand by an even number; then tell the person to add the two products together and tell you if the sum total be odd or even. If the sum be even, the even number is in the right hand, and if it be odd the even number is in the left hand.

— Miscellaneous Notes and Queries, January 1892

… are all prime.

In the 1950s, physicists George Gamow and Moritz Stern worked in the same seven-story building. Gamow, on the second floor, noticed that the first elevator to arrive at his office was most often going down. For Stern, on the sixth floor, the first elevator was most often going up. It was as if elves were manufacturing elevator cars in the middle of the building.

You can observe the same phenomenon in most tall buildings, and there are no elves involved. Do you see why it occurs?

5 is one number.

2 and 3 are 5.

Therefore 2 and 3 are one number.

175 = 1¹ + 7² + 5³