This pouch contains two counters. Each is either black or white. Without opening the pouch, prove that one is black and one white.

The probability that the pouch contains two black counters is 1/4; a black and white counter 1/2; and two white counters 1/4.

The chance of drawing one black counter in each case is 1, 1/2, or 0.

So if we combine these values we’ll get the probability of drawing one black counter from the pouch:

(1/4 × 1) + (1/2 × 1/2) + (1/4 × 0) = 1/2

And if the probability of drawing a black counter is 1/2, then the pouch must contain one black and one white counter. Q.E.D.

(J.A.H. Hunter, after Lewis Carroll)

Pick any number in this array and cross off all the other numbers in its row and column. Repeat this four more times, after which no more numbers will be available.

Add your five numbers. If they total exactly 100, you must rob 10 banks and pass the money to me through a series of prearranged drops. Contact me for details.

“Truth does not exist” is an assertion of truth.

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?