If Brown hopes to throw a six in a game of dice and succeeds, we wouldn’t say he threw the six intentionally. If Brown puts his last cartridge into a six-chambered revolver, spins the chamber as he aims it at Smith, his archenemy, pulls the trigger, and kills Smith, we’d say he killed him intentionally. Does that make sense? In both cases Brown hoped for a certain result, in both cases the probability of that result was the same. If Brown didn’t intentionally throw a six, why did he intentionally shoot Smith?

— Leo Katz, Bad Acts and Guilty Minds, 1987


  • What time is it on the sun?
  • PATERNAL, PARENTAL, and PRENATAL are anagrams.
  • If forecastle is pronounced “fo’c’sle,” should forecast be pronounced “folks”?
  • A clock’s second hand is its third hand.
  • “The religion of one seems madness unto another.” — Thomas Browne

Bonus poser: In what sport does only the winning team travel backward?

Thin Thinking

Some of the figures (particularly the holy ones) in El Greco paintings seem unnaturally tall and thin. An ophthalmologist surmised that the painter had a defect of vision that caused him to see people this way.

The zoologist Sir Peter Medawar pointed out that we can reject this conjecture on purely logical grounds. What was his insight?

Click for Answer

“The Unprovable Liar”

‘What I am saying cannot be proved.’

Suppose this statement can be proved. Then what it says must be true. But it says it cannot be proved. If we assume it can be proved, we prove it cannot be proved. So our supposition that it was provable is wrong. With that road closed to us, let’s try the only other one available — let’s suppose it cannot be proved. Since that is precisely what it says, then it is true after all. And this ends our proof of the above statement!

— Gary Hayden and Michael Picard, This Book Does Not Exist, 2009


A letter from Lewis Carroll to 14-year-old Wilton Rix:

Honoured Sir,

Understanding you to be a distinguished algebraist (i.e. distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much.

If x and y are each equal to ‘1,’ it is plain that 2 × (x2y2) = 0, and also that 5 × (xy) = 0.

Hence 2 × (x2y2) = 5 × (xy).

Now divide each side of this equation by (xy).

Then 2 × (x + y) = 5.

But (x + y) = (1 + 1), i.e. = 2.

So that 2 × 2 = 5.

Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day.

I trust you will pity me and will kindly explain the difficulty to

Your obliged, Lewis Carroll


“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

Run Toppers

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.