In December 2001, mathematician Bill Richardson received a call from a television producer asking for a formula that would reveal the total number of gifts given for any specified day in the song “The Twelve Days of Christmas.” He worked out that the total number of presents given on each day is a triangular number:

1
1 + 2
1 + 2 + 3
1 + 2 + 3 + 4

And so the cumulative total is the sum of triangular numbers:

1
1 + 3
1 + 3 + 6
1 + 3 + 6 + 10

So if P_n is the total number of presents given in the first n days, then

which works out to

“which is an elegant result,” Richardson writes. “So, the lucky girl received 1/6 × 12 × 13 × 14 = 364 presents in total. She should have had a very happy Christmas!”

(Bill Richardson, “The Twelve Days of Christmas,” Mathematical Gazette 86:507 [November 2002], 468.)

Seeing the earth from a distance has changed my perception of the solar system as well. Ever since Copernicus’ theory (that the earth was a satellite of the sun, instead of vice versa) gained wide acceptance, men have considered it an irrefutable truth; yet I submit that we still cling emotionally to the pre-Copernican, or Ptolemaic, notion that the earth is the center of everything. The sun comes up at dawn and goes down at dusk, right? Or as the radio commercial describes sunset: ‘When the sun just goes away from the sky …’ Baloney. The sun doesn’t rise or fall: it doesn’t move, it just sits there, and we rotate in front of it. … Everyone knows that, but I really see it now. No longer do I drive down a highway and wish the blinding sun would set; instead I wish we could speed up our rotation a bit and wing around into the shadows more quickly. I do not have to force myself to call this image to mind; it is there, and occasionally, I use it for other things, although admittedly I have to stretch a bit. ‘What a pretty day’ makes me think that it’s always a pretty day somewhere; if not here, then we just happen to be standing in the wrong place. ‘My watch is fast’ translated into no, it’s not, it’s just that you should be standing farther to the east.

— Astronaut Michael Collins, Carrying the Fire, 1975

If a lion hunts a herd of antelope, what rules govern the herd’s behavior? One intriguing possibility is known as selfish herd theory: Rather than acting to benefit the group as a whole, each member positions itself so that there’s at least one other animal between it and the predator. This produces a pattern known as a Voronoi tesselation — if each dot in the diagram above is an antelope, then the surrounding colored region is the area that’s closer to that antelope than to any of its neighbors. If a lion enters your cell, then you’re the antelope that’s going to get eaten.

This understanding helps to explain some herd behavior. Each animal wants to make its “domain of danger” as small as possible and to be as far as possible from the predator. Dominant animals tend to get prime positions near the center, subordinate animals get pushed to the fringes, and the whole formation evolves continuously as predator and prey move about.

Studies have shown that groups of fiddler crabs tend to take up Voronoi patterns fairly quickly when a predator first appears, and to huddle together when the danger increases as each tries to reduce its surrounding polygon. This actually leads some to move toward the predator as they try to reach the center and put others between the hunter and themselves. Those that violate the movement rules tend to get picked off, which reinforces the evolutionary strength of the strategy.

A family of sets is said to be union-closed if, given any two sets in the family, their union is as well. Here’s an example:

{}, {1,3}, {2}, {1,2,3}, {1,2,3,4}

Combine any two of those sets and you’ll get a member of the same family.

Now, provided our family is finite and consists of more than just the empty set, is there always an element that’s present in at least half of the sets? In the example above, the answer is yes: The element 2 appears in three of the five sets.

Will this always be the case? Strange to say, no one knows. Though the problem is almost absurdly simple to state, it has remained unsolved since Péter Frankl first posed it in 1979.

Henning Bruhn and Oliver Schaudt survey the state of the inquiry here. “The union-closed sets conjecture still has a bit of a journey ahead of it,” they conclude. “We hope it will be an exciting trip.”

(Thanks, Drake.)

Here’s the opening of Alice’s Adventures in Wonderland:

Alice was beginning to get very tired of sitting by her sister on the bank, and of having nothing to do; once or twice she had peeped into the book her sister was reading.

Choose any of the first 12 words and tap your way forward in the sentence from that point, tapping one word for each letter. So, for example, if you choose the word Alice, which has five letters, you’d tap was, beginning, to, get, and land on very. Then do the same thing with that word, advancing four letters to land on by. If you keep this up you’ll always arrive at the word sister.

That’s from Martin Gardner; the same trick works with “Twinkle, Twinkle, Little Star,” the opening of the Bible, and countless other texts.

It’s less surprising than it seems — it’s based on a principle called the Kruskal count, proffered originally by Rutgers physicist Martin Kruskal as a card trick. In each case various tributaries merge into a common stream that arrives at a predictable destination. Here’s an analysis (PDF).

Mark Levi notes an interesting coincidence in Why Cats Land on Their Feet (2012): Dividing normal human body temperature (in Celsius) into 100 approximates e:

“The estimate will be on the low side if you run a fever, or on the high side if you have hypothermia,” Levi writes. “This observation makes the natural logarithm — the one with the base e — seem even more natural.”

When David Huffman died in 1999, the world lost a talented computer scientist — Huffman was best known for discovering the Huffman coding technique used in data compression.

But it also lost a pioneer in mathematical origami, an extension of the traditional art of paper folding that applies computational geometry, number theory, coding theory, and linear algebra. The field today is finding wide application, helping researchers to fold everything from proteins to automobile airbags and space-based telescopes.

Huffman was drawn to the work through his investigations into the mathematical properties of “zero curvature” surfaces, studying how paper behaves near creases and apices of cones. During the last two decades of his life he created hundreds of beautiful, perplexing paper models in which the creases were curved rather than straight.

But he kept his folding research largely to himself. He published only one paper on the subject (PDF), and much of what he discovered was lost at his death. “He anticipated a great deal of what other people have since rediscovered or are only now discovering,” laser physicist Robert Lang told the New York Times in 2004. “At least half of what he did is unlike anything I’ve seen.” MIT computer scientist Erik Demaine is working now with Huffman’s family to recover and document his discoveries (PDF).

“I don’t claim to be an artist. I’m not even sure how to define art,” Huffman told an audience in 1979. “But I find it natural that the elegant mathematical theorems associated with paper surfaces should lead to visual elegance as well.”

Here’s another interesting source of complementary sequences. Take any positive irrational number, say , and call it X. Call its reciprocal Y; in this case , or about 0.7. Add 1 to each of X and Y and we get

1 + X ≈ 2.4

1 + Y ≈ 1.7.

Now make a table of the approximate multiples of 1 + X and 1 + Y:

If we drop the fractional part of each number in the table, we’re left with two complementary sequences — every number 1, 2, 3, … appears in one sequence or the other, but never in both.

They’re called Beatty sequences, after Sam Beatty of the University of Toronto, who discovered them in 1926. A pretty proof by A. Ostrowski and J. Hyslop appears in the March 1927 issue of the American Mathematical Monthly and in Ross Honsberger’s Ingenuity in Mathematics (1970).

In 1934, Indian mathematician S.P. Sundaram proposed this “sieve” for finding prime numbers.

In the first row of a table, write the arithmetic progression 4, 7, 10, …, with the first term 4 and a common difference of 3.

Copy these values into the first column, and then complete each row with its own arithemetic progression, with common differences of 3, 5, 7, 9 …, in successive rows.

Now, remarkably, for any natural number N > 2, if N occurs in the table then 2N + 1 is not a prime number, and if N does not occur in the table, then 2N + 1 is a prime number. (For example, 17 appears in the table, so 35 is not prime; 23 does not appear in the table, so 47 is prime.)

(From Ross Honsberger, Ingenuity in Mathematics, 1970.)