Science & Math

Say Red

https://commons.wikimedia.org/wiki/Category:OpenClipart_ornamental_playing_cards

Cornell mathematician Robert Connelly devised this intuition-defying card game. I shuffle a standard deck of 52 cards and deal them out in a row before you, one at a time. At some point before the last card is dealt, you must say the word “red.” If the next card I deal is red, you win $1; if it’s black you lose $1. If you play blind, your chance of winning is 1/2. Can you improve on this by devising a strategy that considers the dealt cards?

Surprisingly, the answer is no. Imagine a deck with two red cards and two black. Now there are six equally likely deals:

RRBB
RBBR
BBRR
RBRB
BRBR
BRRB

By counting, we can see that the chance of success remains 1/2 regardless of whether you call red before the first, second, third, or fourth card.

Trying to outsmart the cards doesn’t help. You might resolve to wait and see the first card: If it’s black you’ll call red immediately, and if it’s red you’ll wait until the fourth card. It’s true that this strategy gives you a 2/3 chance of winning if the first card is black — but if it’s red then it has a 2/3 chance of losing.

Similarly, it would seem that if the first two cards are black then you have a sure thing — the next card must be red. This is true, but it will happen only once in six deals; on the other five deals, calling red at the third card wins only 2/5 of the time — so this strategy has an overall success rate of (1/6 × 1) + (5/6 × 2/5) = 1/2, just like the others. The cards conspire to erase every seeming advantage.

The same principle holds for a 52-card deck, or indeed for any deck. In general, if a deck has r red cards and b black ones, then your chance of winning, by any strategy whatsoever, is r/(b + r). Seeing the cards that have already been dealt, surprisingly, is no advantage.

(Robert Connelly, “Say Red,” Pallbearers Review 9 [1974], 702.)

Six by Six

The sestina is an unusual form of poetry: Each of its six stanzas uses the same six line-ending words, rotated according to a set pattern:

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This intriguingly insistent form has appealed to verse writers since the 12th century. “In a good sestina the poet has six words, six images, six ideas so urgently in his mind that he cannot get away from them,” wrote John Frederick Nims. “He wants to test them in all possible combinations and come to a conclusion about their relationship.”

But the pattern of permutation also intrigues mathematicians. “It is a mathematical property of any permutation of 1, 2, 3, 4, 5, 6 that when it is repeatedly combined with itself, all of the numbers will return to their original positions after six or fewer iterations,” writes Robert Tubbs in Mathematics in Twentieth-Century Literature and Art. “The question is, are there other permutations of 1, 2, 3, 4, 5, 6 that have the property that after six iterations, and not before, all of the numbers will be back in their original positions? The answer is that there are many — there are 120 such permutations. We will probably never know the aesthetic reason poets settled on the above permutation to structure the classical sestina.”

In 1986 the members of the French experimental writers’ workshop Oulipo began to apply group theory to plumb the possibilities of the form, and in 2007 Pacific University mathematician Caleb Emmons offered the ultimate hat trick: A mathematical proof about sestinas written as a sestina:

emmons sestina

Bonus: When not doing math and poetry, Emmons runs the Journal of Universal Rejection, which promises to reject every paper it receives: “Reprobatio certa, hora incerta.”

(Caleb Emmons, “S|{e,s,t,i,n,a}|“, The Mathematical Intelligencer, December 2007.) (Thanks, Robert and Kat.)

Overheard

A puzzle by Princeton mathematician John Horton Conway:

Last night I sat behind two wizards on a bus, and overheard the following:

A: I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.

B: How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?

A: No.

B: Aha! AT LAST I know how old you are!

“This is an incredible puzzle,” writes MIT research affiliate Tanya Khovanova. “This is also an underappreciated puzzle. It is more interesting than it might seem. When someone announces the answer, it is not clear whether they have solved it completely.”

We can start by auditioning various bus numbers. For example, the number of the bus cannot have been 5, because in each possible case the wizard’s age and the number of his children would then uniquely determine their ages — if the wizard is 3 years old and has 3 children, then their ages must be 1, 1, and 3 and he cannot have said “No.” So the bus number cannot be 5.

As we work our way into higher bus numbers this uniqueness disappears, but it’s replaced by another problem — the second wizard must be able to deduce the first wizard’s age despite the ambiguity. For example, if the bus number is 21 and the first wizard tells us that he’s 96 years old and has three children, then it’s true that we can’t work out the children’s ages: They might be 1, 8, and 12 or 2, 3, and 16. But when the wizard informs us of this, we can’t declare triumphantly that at last we know how old he is, because we don’t — he might be 96, but he might also be 240, with children aged 4, 5, and 12 or 3, 8, and 10. So the dialogue above cannot have taken place.

But notice that if we increase the bus number by 1, to 22, then all the math above will still work if we give the wizard an extra 1-year-old child: He might now be 96 years old with four children ages 1, 1, 8, and 12 or 1, 2, 3, and 16; or he might be 240 with four children ages 1, 4, 5, and 12 or 1, 3, 8, and 10. The number of children increases by 1, the sum of their ages increases by 1, and the product remains the same. So if bus number b produces two possible ages for Wizard A, then so will bus number b + 1 — which means that we don’t have to check any bus numbers larger than 21.

This limits the problem to a manageable size, and it turns out that the bus number is 12 and Wizard A is 48 — that’s the only age for which the bus number and the number of children do not uniquely determine the children’s ages (they might be 2, 2, 2, and 6 or 1, 3, 4, and 4).

(Tanya Khovanova, “Conway’s Wizards,” The Mathematical Intelligencer, December 2013.)

Two similar puzzles: A Curious Conversation and A Curious Exchange.

Sweet Science

monod flan recipe

Sahara geology presented as a flan recipe, from French naturalist Théodore Monod’s Méharées: Explorations au vrai Sahara, 1937:

Take a flan-tray, which represents the basement (our Mauretanian and Tuareg granites).

  1. Place some pastry in the flan-tray in irregular masses (A) — these are the Precambrian mountain chains, the Saharides.
  2. Level this off with a knife (B) so that the folds, as in erosional peneplanation of the Sahara, are seen only in the ravines which cross the plain; the mountains are now vigorously planed down.
  3. First event: a tap (from which, fortunately, jam flows) floods the garnished mould (C). Similarly the sea at the beginning of the Palaeozoic invaded the Saharan basement, which it then partly occupied, until the middle Carboniferous — what an enormous amount of jam! All this time the Sahara is under water, and sandstones, limestones, conglomerates and shales were deposited — all the sediments of the Tuareg and Mauritanian plateaux.
  4. A new event (the djinns must have been at work here) — the bottom of the flan-tray experiences an uplift; the dish, pastry and jam emerge (D). This is the time of the coal measures; the sea retreats, and the Sahara is left high and dry, basking in the sun.
  5. But whoever says dry land, implies erosion; the sediments rise up, are corroded, and the spoon cuts so deeply that it exposes the jam, pastry, and sometimes even the metal of the flan-tray (E).
  6. And while this continues for millions of years, erosion is unable to evaporate its own debris and the eroded sediments are not washed away to the sea — they just accumulate, and what is lost in some districts is gained by others, whilst gradual infilling continues (F).
  7. Then one fine day, while iguanodons are blundering around in Picardy, and swarms of ammonites are scudding around in the Parisian sea, a second tap is turned on again and adds another layer, this time of cream (for convenience of explanation) (G). The sea re-invades a good part of the Sahara and deposits the usual sediments — Cretaceous and Eocene.
  8. A new retreat of the sea and a new continental phase occur, with customary erosion and deposition (H).
  9. Gradually, the country comes to be like it is today; sprinkle with granular sugar (fresh-water Quaternary deposits), and icing sugar dunes (I).
  10. And there we are! Serve hot or chilled.

“Very well — that will teach me to invent foolish nonsense for my neophyte when it is so easy to explain the influence of Saharidian tectonics on the orientation of Hercynian virgations, the suggestion of angular discordnce separating the basal congomerate of the continental beds from the post-Visean argillites, or more simply the origin of the bowlingite included in the pigeonite andesite with diabase facies of Telig. But I doubt that he would understand it any better …”

The Paradox of Goals

Suppose that two teams of equal ability are playing football. If goals are scored at regular intervals, it seems natural to expect that each team will be in the lead for half the playing time. Surprisingly, this isn’t so: If a total of n = 20 goals are scored, then the probability that Team A leads after the first 10 goals and Team B leads after the second 10 goals is only 6 percent, while the probability that one team leads throughout the entire game is about 35 percent. (When the scores are equal, the leading team is considered to be the one that was leading before the last goal.) And the chance that one team leads throughout the second half is 50 percent, no matter how large n is.

Such questions began with a study of ballot problems: In 1887 Joseph Bertrand found that if in an election Candidate P scores p votes and Candidate Q scores q votes, where p > q, then the probability that P leads throughout the voting is (pq)/(p + q).

But pursuing them has led to “conclusions that play havoc with our intuition,” writes Princeton mathematician William Feller. If Peter and Paul toss a coin 20,000 times, we tend to think that each will lead about half the time. But in fact it is 88 times more probable that Peter leads in all 20,000 trials than that each player leads in 10,000 trials. No matter how long the series of coin tosses runs, the most probable number of changes of lead is zero.

“In short, if a modern educator or psychologist were to describe the long-run case histories of individual coin-tossing games, he would classify the majority of coins as maladjusted,” Feller writes. “If many coins are tossed n times each, a surprisingly large proportion of them will leave one player in the lead almost all the time; and in very few cases will the lead change sides and fluctuate in the manner that is generally expected of a well-behaved coin.”

(Gábor J. Székely, Paradoxes in Probability Theory and Mathematical Statistics, 2001; William Feller, An Introduction to Probability Theory and Its Applications, 1957.)

Curves of Constant Width

Trap a circle inside a square and it can turn happily in its prison — a circle has the same breadth in any orientation.

Perhaps surprisingly, circles are not the only shapes with this property. The Reuleaux triangle has the same width in any orientation, so it can perform the same trick:

In fact any square can accommodate a whole range of “curves of constant width,” all of which have the same perimeter (πd, like the circle). Some of these are surprisingly familiar: The heptagonal British 20p and 50p coins and the 11-sided Canadian dollar coin have constant widths so that vending machines can recognize them. What other applications are possible? In the June 2014 issue of the Mathematical Intelligencer, Monash University mathematician Burkard Polster notes that a curve of constant width can produce a bit that drills square holes:

… and a unicycle with bewitching wheels:

The self-accommodating nature of such shapes permits them to take part in fascinating “dances,” such as this one among seven triangles:

This inspired Kenichi Miura to propose a water wheel whose buckets are Reuleaux triangles. As the wheel turns, each pair of adjacent buckets touch at a single point, so that no water is lost:

Here’s an immediately practical application: Retired Chinese military officer Guan Baihua has designed a bicycle with non-circular wheels of constant width — the rider’s weight rests on top of the wheels and the suspension accommodates the shifting axles:

(Burkard Polster, “Kenichi Miura’s Water Wheel, or the Dance of the Shapes of Constant Width,” Mathematical Intelligencer, June 2014.)

Holdouts

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Only three countries have not officially adopted the metric system: Liberia, Myanmar, and the United States.

In October 2013 Myanmar announced that it plans to make the switch.

Chinese Magic Mirrors

During China’s Han dynasty, artisans began casting solid bronze mirrors with a perplexing property. The front of each mirror was a polished, reflective surface, and the back featured a design that had been cast into the bronze. But if light were cast from the mirrored side onto a wall, the design would appear there as if by magic.

The mirrors first came to the attention of the West in the early 19th century, and their secret eluded investigators for 100 years until British physicist William Bragg worked it out in 1932. Each mirror had been cast flat with the design on the reverse side, giving the disk a varying thickness. As the front was polished to produce a convex mirror, the thinner parts of the disk bulged outward slightly. These imperfections are invisible to direct inspection; as Bragg wrote, “Only the magnifying effect of reflection makes them plain.”

Joseph Needham, the historian of ancient Chinese science, calls this “the first step on the road to knowledge about the minute structure of metal surfaces.”

Turing’s Paintbrush

aaron's garden

Shortly after joining the faculty of UC San Diego in 1968, British artist Harold Cohen asked, “What are the minimum conditions under which a set of marks functions as an image?” He set out to answer this by writing a computer program that would create original artistic images.

The result, which he dubbed AARON, has been drawing new images since 1973, first still lifes, then people, then full interior scenes with color. These have been exhibited in galleries throughout the world.

Carnegie Mellon philosopher David E. Carrier writes, “A majority of the viewers of AARON’s work find recognizable shapes in it; the drawing above appears to contain human figures. But AARON here used only the twenty or thirty rules it usually uses, with no special reference to human beings. Does knowing this tell us something about the structure of representation?”

Cohen asks, “If what AARON is making is not art, what is it exactly, and in what ways, other than its origin, does it differ from the ‘real thing?’ If it is not thinking, what exactly is it doing?”

“At the risk of stating the obvious, it seems to me that one of the things human beings find interesting about drawings in general is that they are made by other human beings, and here you are watching the image develop as if it is being developed by another human being. … When the drawing is finished, it functions as a human drawing. … A large part of what we value in art is not the ability of the artist to communicate special meanings, but rather the ability of the artist to present the viewer with something that stimulates the viewer’s own propensity to generate meaning.”

A Tidy Theorem

If an equilateral triangle is inscribed in a circle, then the distance from any point on the circle to the triangle’s farthest vertex is equal to the sum of its distances to the two nearer vertices (above, q = p + r).

(A corollary of Ptolemy’s theorem.)

Sad Magic

sallows tragic square

The magic square at upper left arranges the numbers 3-11 so that each row, column, and long diagonal totals 21.

Lee Sallows found nine tragic words that vary in length from 3 to 11 letters and arranged them into the same square — and he found a unique shape for each word so that every triplet can be assembled into the same 3×7 shape, shown in the border.

Team Spirit

http://commons.wikimedia.org/wiki/File:Hu_Shih_1960_color.jpg

Thomas Huxley’s Evolution and Ethics took China by storm — phrases such as the strong are victorious and the weak perish resonated in the national consciousness and “spread like a prairie fire, setting ablaze the hearts and blood of many young people,” noted philosopher Hu Shih.

People even adopted Darwin’s ideas as names. “The once famous General Chen Chiung-ming called himself ‘Ching-tsun’ or ‘Struggling for Existence.’ Two of my schoolmates bore the names ‘Natural Selection Yang’ and ‘Struggle for Existence Sun.’

“Even my own name bears witness to the great vogue of evolutionism in China. I remember distinctly the morning when I asked my second brother to suggest a literary name for me. After only a moment’s reflection, he said, ‘How about the word shih [fitness] in the phrase “Survival of the Fittest”?’ I agreed and, first using it as a nom de plume, finally adopted it in 1910 as my name.”

(Hu Shih, Living Philosophies, 1931.)

The Pythagoras Paradox

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Draw a right triangle whose legs a and b each measure 1. Draw d and e to complete a unit square. Clearly d + e = 2.

Now if we cut a “step” into the square as shown, then f + h = 1 and g + i = 1, so the total length of the “staircase” is still 2. Cut still finer steps and j + k + l + m + n + o + p + q is likewise 2.

And so on: The more finely we cut the steps, the more closely their shape approximates that of the original triangle’s diagonal. Yet the total length of the stairstep shape remains 2, the sum of its horizontal and vertical elements. At the limit, then, it would seem that c must measure 2 … but we know that the length of a unit square’s diagonal is the square root of 2. Where is the error?

(Thanks, Alex.)

Round Numbers

A curiosity attributed to a Professor E. Ducci in the 1930s:

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Arrange four nonnegative integers in a circle, as above. Now construct further “cyclic quadruples” of integers by subtracting consecutive pairs, always subtracting the smaller number from the larger. So the quadruple above would produce 22, 8, 38, 8, then 14, 30, 30, 14, and so on.

Ducci found that eventually four equal numbers will occur.

A proof appears in Ross Honsberger’s Ingenuity in Mathematics (1970).

Turn, Turn, Turn

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Image: Flickr

The Hoover Dam contains a star map depicting the sky of the Northern Hemisphere as it appeared at the moment that Franklin Roosevelt dedicated the dam. Artist Oskar Hansen imagined that the massive structure might outlive our civilization, and that the map could help future astronomers to calculate the date of its creation. The center star on the map, Alcyone, is the brightest star in the Pleiades, and our sun occupies a position at the center of a flagpole. The whole map traces a complete sidereal revolution of the equinox, a period of 25,694 of our years, and marks the point of the dam’s dedication in that period.

“Man has always sought to express and preserve the magnitude of his exploits in symbols,” Hansen said in 1935. “The written words are symbols arranged so as to preserve in objectified form the thought of man and to record his variant states, both mental and physical. All other arts are similar as to their symbolic significance. They take their place among the category of human endeavor simply as the interpreter of life to itself. They serve as an outer object typifying the inner process. They form the connecting link between the spiritual and the material world. They are the shadows cast by the realities of the soul.”

Misc

  • Juneau, Alaska, is larger than Rhode Island.
  • After reading Coleridge’s Biographia Literaria, Byron said, “I wish he would explain his explanation.”
  • If A + B + C = 180°, then tan A + tan B + tan C = (tan A)(tan B)(tan C).
  • Five counties meet in the middle of Lake Okeechobee.
  • “Life resembles a novel more often than novels resemble life.” — George Sand

No one knows whether Andrew Jackson was born in North Carolina or South Carolina. The border hadn’t been surveyed well at the time.

Scoop

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When Glenn Seaborg appeared as a guest scientist on the children’s radio show Quiz Kids in 1945, one of the children asked whether any new elements, other than plutonium and neptunium, had been discovered at the Metallurgical Laboratory in Chicago during the war.

In fact two had — Seaborg announced for the first time anywhere that two new elements, with atomic numbers 95 and 96 (americium and curium), had been discovered. He said, “So now you’ll have to tell your teachers to change the 92 elements in your schoolbook to 96 elements.”

In his 1979 Priestley Medal address, Seaborg recalled that many students apparently did bring this knowledge to school. And “judging from some of the letters I received from such youngsters, they were not entirely successful in convincing their teachers.”

Triplets

lee sallows triangle theorem

A pretty new theorem by Lee Sallows: Connect each vertex of a triangle to the midpoint of the opposite side, and place a hinge at that point. Now rotate the smaller triangles about these hinges and you’ll produce three congruent triangles.

If the original triangle is isosceles (or equilateral), then the three resulting triangles will be too.

The theorem appears in the December 2014 issue of Mathematics Magazine.

The Trojan Fly

Achilles overtakes the tortoise and runs on into the sunset, exulting. As he does so, a fly leaves the tortoise’s back, flies to Achilles, then returns to the tortoise, and continues to oscillate between the two as the distance between them grows, changing direction instantaneously each time. Suppose the tortoise travels at 1 mph, Achilles at 5 mph, and the fly at 10 mph. An hour later, where is the fly, and which way is it facing?

Strangely, the fly can be anywhere between the two, facing in either direction. We can find the answer by running the scenario backward, letting the three participants reverse their motions until all three are again abreast. The right answer is the one that returns the fly to the tortoise’s back just as Achilles passes it. But all solutions do this: Place the fly anywhere between Achilles and the tortoise, run the race backward, and the fly will arrive satisfactorily on the tortoise’s back at just the right moment.

This is puzzling. The conditions of the problem allow us to predict exactly where Achilles and the tortoise will be after an hour’s running. But the fly’s position admits of an infinite number of solutions. Why?

(From University of Arizona philosopher Wesley Salmon’s Space, Time, and Motion, after an idea by A.K. Austin.)

Traffic Waves

In 2008, physicist Yuki Sugiyama of the University of Nagoya demonstrated why traffic jams sometimes form in the absence of a bottleneck. He spaced 22 drivers around a 230-meter track and asked them to proceed as steadily as possible at 30 kph, each maintaining a safe distance from the car ahead of it. Because the cars were packed quite densely, irregularities began to appear within a couple of laps. When drivers were forced to brake, they would sometimes overcompensate slightly, forcing the drivers behind them to overcompensate as well. A “stop-and-go wave” developed: A car arriving at the back of the jam was forced to slow down, and one reaching the front could accelerate again to normal speed, producing a living wave that crept backward around the track.

Interestingly, Sugiyama found that this phenomenon arises predictably in the real world. Measurements on various motorways in Germany and Japan have shown that free-flowing traffic becomes congested when the density of cars reaches 40 vehicles per mile. Beyond that point, the flow becomes unstable and stop-and-go waves appear. Because it’s founded in human reaction times, this happens regardless of the country or the speed limit. And as long as the total number of cars on the motorway doesn’t change, the wave rolls backward at a predictable 12 mph.

“Understanding things like traffic jams from a physical point of view is a totally new, emerging field of physics,” Sugiyama told Gavin Pretor-Pinney for The Wavewatcher’s Companion. “While the phenomenon of a jam is so familiar to us, it is still too difficult to truly understand why it happens.”

The Facts

“Boarding-House Geometry,” by Stephen Leacock:

Definitions and Axioms

All boarding-houses are the same boarding-house.
Boarders in the same boardinghouse and on the same flat are equal to one another.
A single room is that which has no parts and no magnitude.
The landlady of a boarding-house is a parallelogram — that is, an oblong angular figure, which cannot be described, but which is equal to anything.
A wrangle is the disinclination of two boarders to each other that meet together but are not in the same line.
All the other rooms being taken, a single room is said to be a double room.

Postulates and Propositions

A pie may be produced any number of times.
The landlady can be reduced to her lowest terms by a series of propositions.
A bee line may be made from any boarding-house to any other boarding-house.
The clothes of a boarding-house bed, though produced ever so far both ways, will not meet.
Any two meals at a boarding-house are together less than two square meals.
If from the opposite ends of a boarding-house a line be drawn passing through all the rooms in turn, then the stovepipe which warms the boarders will lie within that line.
On the same bill and on the same side of it there should not be two charges for the same thing.
If there be two boarders on the same flat, and the amount of side of the one be equal to the amount of side of the other, each to each, and the wrangle between one boarder and the landlady be equal to the wrangle between the landlady and the other, then shall the weekly bills of the two boarders be equal also, each to each.
For if not, let one bill be the greater. Then the other bill is less than it might have been — which is absurd.

From his Literary Lapses, 1918. See Special Projects.

A Late Contribution

A ghost co-authored a mathematics paper in 1990. When Pierre Cartier edited a Festschrift in honor of Alexander Grothendieck’s 60th birthday, Robert Thomas contributed an article that was co-signed by his recently deceased friend Thomas Trobaugh. He explained:

The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom’s simulacrum remarked, ‘The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.’ Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K0 obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom’s simulacrum had been so insistent, I knew he wouldn’t let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper. To Tom, I could have explained why he must be listed as a coauthor.

Thomason himself died suddenly five years later of diabetic shock, at age 43. Perhaps the two are working again together somewhere.

(Robert Thomason and Thomas Trobaugh, “Higher Algebraic K-Theory of Schemes and of Derived Categories,” in P. Cartier et al., eds., The Grothendieck Festschrift Volume III, 1990.)

Cross Purposes

ferland crossword grid

The daily New York Times crossword puzzle fills a grid measuring 15×15. The smallest number of clues ever published in a Times puzzle is 52 (on Dec. 23, 2008), and the largest is 86 (on Jan. 21, 2005).

This set Bloomsburg University mathematician Kevin Ferland wondering: What are the theoretical limits? What are the shortest and longest clue lists that can inform a standard 15×15 crossword grid, using the standard structure rules (connectivity, symmetry, and 3-letter words minimum)?

The shortest is straightforward: A blank grid with no black squares will be filled with 30 15-letter words, 15 across and 15 down.

The longest is harder to determine, but after working out a nine-page proof Ferland found that the answer is 96: The largest number of clues that a Times-style crossword will admit is 96, using a grid such as the one above.

In honor of this result, he composed a puzzle using this grid — it appears in the June-July 2014 issue of the American Mathematical Monthly.

(Kevin K. Ferland, “Record Crossword Puzzles,” American Mathematical Monthly 121:6 [June-July 2014], 534-536.)

All for One

A flock of starlings masses near sunset over Gretna Green in Scotland, preparatory to roosting after a day’s foraging. The flock’s shape has a mesmerisingly fluid quality, flowing, stretching, rippling, and merging with itself. Similarly massive flocks form over Rome and over the marshlands of western Denmark, where more than a million migrating starlings form an enormous display known as the “black sun.”

What rules produce this behavior? In the 1970s scientists thought that the birds might be following an electrostatic field produced by the leader. Earler, in the 1930s, one paper even suggested that they use thought transference.

But in 1986 computer graphics expert Craig Reynolds found that he could create a lifelike virtual flock (below) using a surprisingly simple set of rules: direct each bird to avoid crowding nearby flockmates, steer toward the average heading of nearby flockmates, and move toward the center of mass of nearby flockmates.

Studies with real birds seem to bear this out: Under rules like these a flock can react sensitively to a change in direction by any of its members, permitting the whole group to respond efficiently as one organism. “News of a predator’s approach can be communicated rapidly through the flock by whichever of the hundreds of birds on the outside notice it first,” writes Gavin Pretor-Pinney in The Wavewatcher’s Companion. “When under attack by a peregrine falcon, for instance, starling flocks will contract into a ball and then peel away in a ribbon to distract and confuse the predator.”