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

Australian philosopher J.J.C. Smart asks, “In what units is the rate of time flow to be measured? Seconds per — what?”

The cable guy is coming tomorrow between 8 a.m. and 4 p.m. Let’s bet on whether he turns up in the morning or the afternoon.

Both windows are four hours long, so as we sit here today, it seems rational to treat them as equally likely. But suppose you choose the morning. As the clock begins to tick, the morning window will gradually close, making the afternoon seem increasingly preferable. Though your present self regards the two eventualities as equally likely, it seems that your future self won’t. Should that affect your decision today?

(In my experience the guy never turns up at all, so perhaps that solves it.)

Hajek, Alan (2005), “The Cable Guy Paradox,” Analysis 65: 112-19.