Science & Math

Half Right

“Numero deus impare gaudet [the god delights in odd numbers].” — Virgil

“Why is it that we entertain the belief that for every purpose odd numbers are the most effectual?” — Pliny

“This is the third time; I hope good luck lies in odd numbers. … They say there is divinity in odd numbers, either in nativity, chance, or death.” — Shakespeare, The Merry Wives of Windsor

Simple Enough

Before his students arrived for a graduate course in logic, Raymond Smullyan wrote on the blackboard:


Math Notes

division cancellation

Galton’s Paradox

Suppose you flip three fair coins.

Necessarily two will match, and it’s an even chance whether the third will be head or tail.

Therefore the chance that all three will match is 1/2.

Math Notes

8589934592 × 116415321826934814453125 = 1000000000000000000000000000000000


What’s the difference between six dozen dozen and half a dozen dozen?

If you answered “nothing,” reconsider.


Syllogisms offered in Lewis Carroll’s 1896 textbook in symbolic logic:

1. Babies are illogical.
2. Nobody is despised who can manage a crocodile.
3. Illogical persons are despised.
Therefore babies cannot manage crocodiles.

1. No interesting poems are unpopular among people of real taste.
2. No modern poetry is free from affectation.
3. All your poems are on the subject of soap bubbles.
4. No affected poetry is popular among people of taste.
5. Only a modern poem would be on the subject of soap bubbles.
Therefore all your poems are uninteresting.


Raymond Smullyan proposes a scene in which two men are regarding a blackboard. On the board is written ONLY AN IDIOT WOULD BELIEVE THIS SENTENCE.

The first man says, “Do you believe that sentence?”

The second says, “Of course not. Only an idiot would believe that sentence.”

“He clearly does believe it, yet he says he doesn’t believe it,” Smullyan says. “So he’s in the curious position of believing something and also believing that he doesn’t believe it.”

Double Duty

doubly magic square

A doubly magic square: Every row and column adds to 840 and multiplies to 2,058,068,231,856,000.

“To Take a Man’s Waistcoat Off Without Removing His Coat”

The waistcoat should first be unbuttoned in the front, and then the buckle at the back must be unloosed. The operator, standing in front of the person operated upon, should then place his hands underneath the coat at the back, taking hold of the bottom of the waistcoat, at the same time requesting the wearer to extend his arms at full length over his head. Now raise the bottom part of the waistcoat over the head of the wearer (if the waistcoat be tight it will be necessary to force it a little, but this must not be minded so long as the waistcoat is not torn); the waistcoat then will have been brought to the front of the wearer, across his chest. Take the right side bottom-end of the waistcoat, and put it into the arm-hole of the coat at the shoulder, at the same time putting the hand up the sleeve, seizing the end, and drawing it down the sleeve; this action will release one arm-hole of the garment to be removed. The next thing to be done is to pull the waistcoat back again out of the sleeve of the coat, and put the same end of the waistcoat into the left arm-hole of the coat, again putting the hand up the sleeve of the coat as before, and seizing the end of the garment. It may then be drawn quite through the sleeve, and the puzzle is accomplished.

Cassell’s Complete Book of Sports and Pastimes, 1896


In 1610, thinking he had discovered two moons orbiting Saturn, Galileo composed a message:

ALTISSIMUM PLANETAM TERGEMINUM OBSERVAVI (“I have observed the most distant planet to have a triple form”)

… and sent it to Kepler as an anagram:


Remarkably, Kepler managed to “solve” this as a message about Mars, not Saturn:

SALVE UMBISTENEUM GEMINATUM MARTIA PROLES (“Hail, twin companionship, children of Mars”)

The German astronomer had predicted that the Red Planet had two moons, and imagined that Galileo was confirming his belief.

There’s a message in this, somewhere.

Open and Shut

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)

Fortune Square

fortune square

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.

Never Mind

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

Math Notes

math notes

Problem Solved

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.”


Said the chemist, “I’ll take some dimethyloximidomesoralamide
And I’ll add just a dash of dimethylamidoazobensaldehyde;
But if these won’t mix,
I’ll just have to fix
Up a big dose of trisodiumpholoroglucintricarboxycide.”

“Odd or Even”

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

Math Notes

truncated prime

… are all prime.

The Elevator Paradox

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?

More Geometry Trouble

more geometry trouble

The top figure, measuring 8 × 8, can be reassembled to form the bottom figure, measuring 5 × 13. Thus 64 = 65.

Pi in Verse

pi in verse

— A.C. Orr, Literary Digest, 1906

Spud Loops

Given any pair of potatoes — even bizarre, Richard Nixon-shaped potatoes — it’s always possible to draw a loop on each so that the two loops are identical in three dimensions.

Do you see the simple, intuitive proof for this?

Click for Answer


5 is one number.

2 and 3 are 5.

Therefore 2 and 3 are one number.