From Lewis Carroll:

Men over 5 feet high are numerous.

Men over 10 feet high are not numerous.

Therefore men over 10 feet high are not over 5 feet high.

“How are you going to teach logic in a world where everybody talks about the sun setting, when it’s really the horizon rising?” — Cal Craig, quoted in Howard Eves, Mathematical Circles Revisited, 1971

Let’s play a game. We’ll each name three consecutive outcomes of a coin toss (for example, tails-heads-heads, or THH). Then we’ll flip a coin repeatedly until one of our chosen runs appears. That player wins.

Is there any strategy you can take to improve your chance of beating me? Strangely, there is. When I’ve named my triplet (say, HTH), take the complement of the center symbol and add it to the beginning, and then discard the last symbol (here yielding HHT). This new triplet will be more likely to appear than mine.

The remarkable thing is that this always works. No matter what triplet I pick, this method will always produce a triplet that is more likely to appear than mine. It was discovered by Barry Wolk of the University of Manitoba, building on a discovery by Walter Penney.

It’s said that British Astronomer Royal G.B. Airy once discovered an empty box at the Greenwich Observatory in London.

He wrote EMPTY BOX on a piece of paper and put it inside.

“Attached to the outside, such a label is true,” write Gary Hayden and Michael Picard in This Book Does Not Exist. “Placed inside the box, it makes itself false. Alternatively, suppose the label says: ‘The box this label is inside is empty.’ Outside of any box, the subject of this sentence fails to refer — there is no box inside which the label is located. However, once inside an otherwise empty box, the sentence becomes false.”

A placebo has no pharmaceutical properties; if it works, it works only because of my own belief in its efficacy.

If I know that I’m taking a placebo, it will be ineffective.

So while the placebo cures me only because I believe it will, I can’t believe that it will cure me only because I believe it will.

(From City University philosopher Peter Cave.)

A paradox by Sam Loyd. The figure on the left, measuring 8×8, can be reassembled into the figure on the right, measuring 7×9. Therefore 64 = 63.

A set of dominoes can be arranged into a valid arithmetic sum:

and into a magic square:

From Joseph S. Madachy, Madachy’s Mathematical Recreations, 1966, and W.W. Rouse Ball, Mathematical Recreations and Essays, 1919.

I fire two shots and kill Cristabel. The first bullet strikes her brain, killing her immediately. The second bullet lodges in her heart: it would have killed her, had she not already died because of the first bullet. I argue that I did no serious harm. Bearing in mind what the second bullet would have done, the first bullet merely caused Cristabel the loss of one second of life — hardly serious. The second bullet, of course, did not kill her.

— Peter Cave, This Sentence Is False, 2009

5³ + 5³ = 250
2³ + 5³ + 0³ = 133
1³ + 3³ + 3³ = 55