Thomas Edison offered this burlesque on perpetual motion. “There will always be a nine opposed to a six,” explains Sam Loyd, “and as nine weighs more than six, it will make the wheel revolve rapidly, as well as your head when you understand it thoroughly.”
A contronym is a word with two contrary meanings, such as cleave or sanction (more here).
The word contronym itself has no double meaning. Is it a contronym?
“Not until I came along!” writes Charles Melton in Word Ways. “I declare that it is a contronym for the simple reason that it isn’t! It is both a self-opposite and not a self-opposite. QED.”
A logic exercise by Lewis Carroll. What conclusion is implied by these premises?
- Animals that do not kick are always unexcitable.
- Donkeys have no horns.
- A buffalo can always toss one over a gate.
- No animals that kick are easy to swallow.
- No hornless animal can toss one over a gate.
- All animals are excitable except buffaloes.
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?
You say you know your brother.
Yet when your brother is hooded you are unable to identify him.
Therefore you both do and do not know your brother.
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?
‘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:
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 × (x2 – y2) = 0, and also that 5 × (x – y) = 0.
Hence 2 × (x2 – y2) = 5 × (x – y).
Now divide each side of this equation by (x – y).
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
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.
- Can one keep a promise unintentionally?
- The plural of u is us.
- 1676 = 11 + 62 + 73 + 64
- DISMANTLEMENT and SKEPTICISM are typed with alternating hands.
- “He was lucky and he knew it.” — Clark Gable’s proposed epitaph
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?”
- Cain killed a quarter of the world’s population.
- Spencer Tracy’s 1937 Oscar was engraved DICK TRACY.
- 15626 = 1 + 56×2-6
- NINE TEN ELEVEN alternates vowels and consonants.
- “Can you play chess without the queen?” — Wittgenstein
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.
This card curiosity is attributed to Lewis Carroll.
Lay down eight cards with these values:
Now add the values in each column, find a card of that value in the deck, and place it on top of the lower card. Aces count as 1, jacks as 11, queens as 12, and kings as 13. Thus in the first column 1 + 2 = 3, so you’d place a 3 on top of the 2.
When you’ve done all four columns, repeat the process, placing a 4 on the 3, etc. If a sum is more than 13, subtract 13 from it (for example, Q + 7 = 19 – 13 = 6). When you’ve exhausted the deck you’ll have four kings in the bottom row. Place each of these piles on the card above it, then take up the packs from right to left.
Turn the deck face down again and deal 13 cards in a circle, making note of which was dealt first. Counting from that card, deal 13 more cards, placing them on every second pile (that is, piles 2, 4, 6, etc.) and continuing around the circle until 13 are dealt. Then deal 13 more cards, one onto each third pile (piles 3, 6, 9, etc.), and finish by dealing the last 13 cards, one onto every fourth pile (4, 8, 12, etc.).
Each pile should now contain four cards. Take them up in order, starting with the first.
Now comes the payoff. Spell aloud A-C-E, dealing a card for each letter and turning the last one face up. It will be an ace. Continue with T-W-O, T-H-R-E-E, and so on up through J-A-C-K, Q-U-E-E-N, and K-I-N-G. In each case the last card will have the rank just spelled — and the full count will precisely exhaust the deck.