Science & Math

Din Minimum

In 1958, acoustician William MacLean of the Polytechnic Institute of Brooklyn answered a perennial question: How many guests can attend a cocktail party before it becomes too noisy for conversation? He declared that the answer, for a given room, is

cocktail party noise


N0 = the critical number of guests above which each speaker will try overcome the background noise by raising his voice
K = the average number of guests in each conversational group
a = the average sound absorption coefficient of the room
V = the room’s volume
h = a properly weighted mean free path of a ray of sound
d0 = the conventional minimum distance between speakers
Sm = the minimum signal-to-noise ratio for the listeners

When the critical guest N0 arrives, each speaker is forced to increase his acoustic power in small increments (“I really don’t know what she sees in him.” — “Beg your pardon?” — “I say, I REALLY DON’T KNOW WHY SHE GOES OUT WITH HIM”) until each group is forced to huddle uncomfortably close in order to continue the conversation.

“We see therefore that, once the critical number of guests is exceeded, the party suddenly becomes a loud one,” MacLean concluded, somewhat sadly. “The power of each talker rises exponentially to a practical maximum, after which each reduces his or her talking distance below the conventional distance and then maintains, servo fashion, just the proximity, tête à tête, required to attain a workable signal-to-noise ratio. Thanks to this phenomenon the party, although a loud one, can still be confined within one apartment.”

(William R. MacLean, “On the Acoustics of Cocktail Parties,” Journal of the Acoustical Society of America, January 1959, 79-80.)

Construction Set

From the ever-inventive Lee Sallows, a self-tiling tile set:

sallows self-tiling tiles

His article on such self-similar tilings appears in the December 2012 issue of Mathematics Magazine.

Vanishing Act

In 1913 mathematician P.E.B. Jourdain proposed a familiar paradox:

On one side of a blank strip of paper, write The statement on the other side of this paper is true.

On the other side, write The statement on the other side of this paper is false.

“The paradox in this form is quite vulnerable to an absolute refutation,” wrote Valdis Augstkalns in a 1970 letter to The Listener. “One takes the paper, gives it a half twist, and joins the ends to form a Möbius strip. The serious and philosophically legitimate question is transformed to ‘Eminent members of the panel, which is the other side of the paper?’”

Topsy Turvy

This reversible magic square comes from Henry Dudeney’s Canterbury Puzzles.

Each row, column, and diagonal in the square totals 179.

Thanks to some clever calligraphy, this remains true when the square is turned upside down.


  • A pound of dimes has the same value as a pound of quarters.
  • The French word hétérogénéité has five accents.
  • 32768 = (3 – 2 + 7)6 / 8
  • Can you deceive yourself deliberately?
  • “My country is the world, and my religion is to do good.” — Thomas Paine

In 2000, Guatemalan police asked Christmas revelers not to fire pistols into the air. “Lots of people die when bullets fall on their heads,” National Civilian Police spokesman Faustino Sanchez told Reuters. He said that five to ten Guatemalans are killed or injured each Christmas by falling bullets.

The Chosen Ones

J.B.S. Haldane was once asked what his study of biology had taught him about God.

He said that the Creator, if he exists, has “an inordinate fondness for beetles.”

Number Squares

Howard W. Bergerson devised these:

number squares

As in a word square, the “columns” in each sum are identical with the “rows.”

Gene Pool

Asked whether he would give his life to save a drowning brother, J.B.S. Haldane said, “No, but I would to save two brothers or eight cousins.”


In 1877, five-year-old Bertrand Russell asked his Aunt Agatha, “Aunty, do limpets think?”

She said, “I don’t know.”

He said, “Then you must learn.”


  • Will Rogers died at the northernmost point in the United States.
  • 94122 + 23532 = 94122353
  • TO BE OR NOT TO BE contains two Bs.
  • If you stop me being mute, what sound do I make?
  • “Better to ask twice than to lose your way once.” — Danish proverb

Inside Out

Erl E. Kepner patented a bewildering object in 2002 — a one-sided coffee mug:

To help us see the unique properties of the beverage vessel, let’s pretend that the vessel is made of astonishingly thin material. This is a ‘thought experiment’, not a real experiment, where you actually physically do anything. Note that the only edge on the vessel is the rim that your lips would touch if you drank coffee from it. The rim of the container would be a very sharp edge while the areas where the container and the hollow handle come together would be smooth curved shapes. Now pretend that you have a very tiny little black ball shaped magnet located on the surface of the vessel somewhere and another little tiny white ball magnet located on the opposite side of the vessel material. If one were to (mentally) move either the black or white ball magnet, it would cause the white or black ball magnet to move also. One could move the black or white ball magnets, one at a time, along the vessel surface so that white ball magnet ended up where the black ball magnet was initially located and the black ball magnet was where the white ball magnet originally was. This can be accomplished without having either of the magnets pass over the vessel rim, which is the only edge of the vessel. Other than the Klein Bottle, no other hollow shape has this property. Another way to envision or demonstrate the unique properties of this shape is to point out the fact that a little bug can crawl from any point on the surface of the vessel to any other point on the surface of the vessel without crossing over an edge. Bugs cannot do this on a normal coffee cup or any other three-dimensional shape that we use in our daily lives.

“The future marketing of the beverage container of this invention will use these sorts of interesting points to stimulate interest among technically well educated people and everyday people with an innate curiosity and appreciation for the wonder and beauty of mathematics and nature.”

See Carry-All and the Klein bottle recycling center.

The Perfect Crime

Suspect A has shot a man through the heart during the last half minute. But Suspect B shot him through the heart during the preceding 1/4 minute, Suspect C shot him through the heart during the 1/8 minute before that, and so on. Assuming that a bullet through the heart kills a man instantly, the victim must already have been dead before any given suspect shot him.

Indeed, notes José Benardete, he cannot be said to have died of a bullet wound.

“The Research Man’s Prayer”

Help me be MANIC so I may be joyous though the results are equivocal.

Help me be DEPRESSIVE for when a prediction is verified, I must know that it will not later be confirmed.

Help me be SADISTIC so I suffer not though the subjects be sorely anguished.

Help me be MASOCHISTIC for even the most obstinate experimental animal should be a pleasure to me.

Help me be PSYCHOPATHIC to quiet the guilt when I tell loved ones that the experiment is going well.

Help me be SCHIZOPHRENIC to sustain myself by finding hopeful trends in random data.

Help me be PARANOID so I can see in the hostile attitudes of others the supremacy of my own work.

Help me to have ANXIETY ATTACKS so that even on holidays I find myself toiling in the laboratory.

And finally,

Help my wife get a job! for when I cross over the shadowy border of normalcy, somebody will have to support the kids. Amen.

– R.A McCleary in the Worm Runner’s Digest, November 1960

Double Trouble

The properties of the simple Möbius strip are well understood: Take a strip of paper, give it a half-twist, and tape the ends together. Now an ant can traverse the full length of the loop, on both sides, and return to its starting point without ever crossing an edge.

But try doing the same thing with two strips of paper. Pair the strips, give them a half-twist, and connect the ends. Now it’s possible to insert a toothpick between the bands and to draw the toothpick along the entire length of the loop, which seems to show that they’re two distinct objects. But if you draw a line along either strip, starting anywhere, you’ll find that you traverse both strips and return to your starting point.

“I have known people to ponder this for hours while listening to Pink Floyd without ever fully appreciating what they have beheld,” writes Clifford Pickover in The Möbius Strip. Are you holding one object or two?

The Paradox of the Second Ace

You’re watching four statisticians play bridge. After a hand is dealt, you choose a player and ask, “Do you have at least one ace?” If she answers yes, the chance that she’s holding more than one ace is 5359/14498, which is less than 37 percent.

On a later hand, you choose a player and ask, “Do you have the ace of spades?” Strangely, if she says yes now the chance that she has more than one ace is 11686/20825, which is more than 56 percent.

Why does specifying the suit of her ace improve the odds that she’s holding more than one ace? Because, though a smaller number of potential hands contain that particular ace, a greater proportion of those hands contain a second ace. It’s counterintuitive, but it’s true.

A Change of Key

5 × 55 × 555 = 152625

remains true if each digit is increased by 1:

6 × 66 × 666 = 263736

Brunnian Links

The standard braid has a curious property: If we remove any one of the three strands, the other two are seen to be unconnected. If we remove the black strand above, the blue and red strands simply snake along one above the other. Similarly, removing the red or the blue strand reveals that the remaining strands are not braided together.

See Borromean Rings.

Number Forms

When thinking of numbers, about 5 percent of the population see them arranged on a sort of mental map. The shape varies from person to person, assuming “all sorts of angles, bends, curves, and zigzags,” in the words of Francis Galton, who described them first in The Visions of Sane Persons (1881). Usually the forms are two-dimensional, but occasionally they twist through space or bear color.

People who have forms report that they remain unchanged throughout life, but having one is such a peculiarly personal experience that “it would seem that a person having even a complicated form might live and die without knowing it, or at least without once fixing his attention upon it or speaking of it to his nearest friends,” wrote philosopher G.T.W. Patrick in 1893. One man told mathematician Underwood Dudley that “when he told his wife about his number form, she looked at him oddly, as if he were unusual, when he thought that she was the peculiar one because she did not have one.”

The phenomenon is poorly understood even today; possibly it arises because of a cross-activation between the parts of the brain that recognize spatial relationships and numbers. Two of Dudley’s students were identical twins; both had forms, but the forms were different. “Although our understanding of how the brain works has advanced since 1880, it probably has not advanced enough to deal with number forms,” he writes. “Another hundred years or so may be needed.”


  • A TOYOTA’S A TOYOTA is a palindrome.
  • Lee Trevino was struck by lightning in 1975.
  • 39343 = 39 + 343
  • “Money often costs too much.” — Emerson

Guy Debord’s 1957 autobiography, Mémoires, was bound in a sandpaper cover so that it would destroy any book placed next to it.

(Thanks, Vinny.)

Soul Support

“It seems to me immensely unlikely that mind is a mere by-product of matter. For if my mental processes are determined wholly by the motions of atoms in my brain I have no reason to suppose that my beliefs are true. They may be sound chemically, but that does not make them sound logically. And hence I have no reason for supposing my brain to be composed of atoms.” — J.B.S. Haldane, Possible Worlds, 1927

Double Talk

A logical curiosity by L.J. Cohen: A policeman testifies that nothing a prisoner says is true, and the prisoner testifies that something the policeman says is true. The policeman’s statement can’t be right, as that leads immediately to a contradiction. This means that something the prisoner says is true — either a new statement or his current one. If it’s a new statement, then we establish that the prisoner says something else. If it’s his current statement, then the policeman must say something else (as we know that his current statement is false).

J.L. Mackie writes, “From the mere fact that each of them says these things — not from their being true — it follows logically, as an interpretation of a formally valid proof, that one of them — either of them — must say something else. And hence, by contraposition, if neither said anything else they logically could not both say what they are supposed to say, though each could say what he is supposed to say so long as the other did not.”

The Devil’s Game

Ms. C dies and goes to hell, where the devil offers a game of chance. If she plays today, she has a 1/2 chance of winning; if she plays tomorrow, the chance will be 2/3; and so on. If she wins, she can go to heaven, but if she loses she must stay in hell forever. When should she play?

The answer is not clear. If she waits a full year, her probability of winning will have risen to about 0.997268. At that point, waiting an additional day will improve her chances by only about 0.000007. But at stake is infinite joy, and 0.000007 multiplied by infinity is infinite. And the additional day spent waiting will contain (presumably) only a finite amount of torment. So it seems that the expected benefit from a further delay will always outweigh the cost.

“This logic might suggest that Ms. C should wait forever, but clearly such a strategy would be self-defeating,” wrote Edward J. Gracely in proposing this conundrum in Analysis in 1988. “Why should she stay forever in a place in order to increase her chances of leaving it? So the question remains: what should Ms. C do?”

The Copernicus Method

Princeton astrophysicist J. Richard Gott was visiting the Berlin Wall in 1969 when a curious thought occurred to him. His visit occurred at a random moment in the wall’s existence. So it seemed reasonable to assume that there was a 50 percent chance that he was observing it in the middle two quarters of its lifetime. “If I was at the beginning of this interval, then one-quarter of the wall’s life had passed and three-quarters remained,” he wrote later in New Scientist. “On the other hand, if I was at the end of of this interval, then three-quarters had passed and only one-quarter lay in the future. In this way I reckoned that there was a 50 per cent chance the wall would last from 1/3 to 3 times as long as it had already.”

At the time, the wall was 8 years old, so Gott concluded that there was a 50 percent chance that it would last more than 2-2/3 years but fewer than 24. The 24 years would have elapsed in 1993. The wall came down in 1989.

Encouraged, Gott applied the same principle to estimate the lifetime of the human race. In an article published in Nature in 1993, he argued that there was a 95 percent chance that our species would survive for between 5,100 and 7.8 million years.

When and whether the method is valid is still a matter of debate among physicists and philosophers. But it’s worth noting that on the day Gott’s paper was published, he used it to predict the longevities of 44 plays and musicals on and off Broadway. His accuracy rate was more than 90 percent.

The Sofa Problem

In 1966, Austrian mathematician Leo Moser asked a pleasingly practical question: If a corridor is 1 meter wide, what’s the largest sofa one could squeeze around a corner?

That was 46 years ago, and it’s still an open question. In 1968 Britain’s John Michael Hammersley showed that a sofa shaped somewhat like a telephone receiver could make the turn even if its area were more than 2 square meters (above). In 1992 Joseph Gerver improved this a bit further, but the world’s tenants await a definitive solution.

Similar problems concern moving ladders and pianos. Perhaps what we need are wider corridors.