These are called Borwein integrals, after David and Jonathan Borwein, the father-and-son mathematicians who first presented them in 2001.

Engineer Hanspeter Schmid writes, “[W]hen this fact was recently verified by a researcher using a computer algebra package, he concluded that there must be a ‘bug’ in the software. It is not a bug, though; this series of integrals really only results in π/2 up to a certain point, and then breaks down. This astonishes most mathematically educated readers, as especially those readers mentally extrapolate the sequence shown above and find it surprising that something fundamental should change when the factor sinc(x/15) is introduced.” He gives a graphic explanation of what’s happening.

(David Borwein and Jonathan M. Borwein, “Some Remarkable Properties of Sinc and Related Integrals,” Ramanujan Journal 5:1 [March 2001], 73–89.) (Thanks, Dan.)

In his 2008 book Impossible?, Julian Havil presents an argument offered in Massachusetts in 1854 contending that foreigners were more likely to be insane than native-born Americans. These figures were offered:

Whole Population
	Insane	Not Insane	Totals
Foreign-Born	625	229375	230000
Native-Born	2007	892669	894676
Totals	2632	1122044	1124676

The probability that a foreign-born person was deemed insane was 625/230000 = 2.7 × 10^-3, and for a native-born person the probability was 2007/894676 = 2.2 × 10^-3, which seems to support the claim.

But we get a different story when we divide the data by social hierarchy, into what were called the pauper and independent classes:

Pauper Class
	Insane	Not Insane	Totals
Foreign-Born	182	9090	9272
Native-Born	250	12513	12763
Totals	432	21603	22035

Independent Class
	Insane	Not Insane	Totals
Foreign-Born	443	220285	220728
Native-Born	1757	880156	881913
Totals	2200	1100441	1102641

In the pauper class the probability of a foreign-born person being deemed insane is 182/9272 = 0.02, which is the same as that for a native-born person (250/12763 = 0.02). And the same is true in the independent class, where both probabilities are 2.0 × 10^-3. Havil writes, “So, if an adjustment is made for the status of the individuals we see that there is no relationship at all between sanity and origin” (an example of Simpson’s paradox).

“Squaring the square” refers to tiling a square with other squares, each with sides of integer length.

In a “perfect” squared square, like the one above, each smaller square is of a different size. The Cambridge University team that first sought perfect squares found a novel way to go about it — they transformed the square tiling into an electrical circuit in which each square is a resistor that connects to its neighbors above and below, and then applied Kirchhoff’s circuit laws to that circuit.

The example below isn’t perfect, but the technique did succeed — the smallest perfect square they found is 69 units on a side.

(Rowland Leonard Brooks, et al., “The Dissection of Rectangles into Squares,” Duke Mathematical Journal 7:1 [1940], 312-340.)

In January 2009, marine ecologists Robert Pitman and John Durban were watching a group of Antarctic killer whales wash a seal off an ice floe when, surprisingly, a pair of humpback whales intervened:

Exposed to lethal attack in the open water, the seal swam frantically toward the humpbacks, seeming to seek shelter, perhaps not even aware that they were living animals. (We have known fur seals in the North Pacific to use our vessel as a refuge against attacking killer whales.)

Just as the seal got to the closest humpback, the huge animal rolled over on its back — and the 400-pound seal was swept up onto the humpback’s chest between its massive flippers. Then, as the killer whales moved in closer, the humpback arched its chest, lifting the seal out of the water. The water rushing off that safe platform started to wash the seal back into the sea, but then the humpback gave the seal a gentle nudge with its flipper, back to the middle of its chest. Moments later the seal scrambled off and swam to the safety of a nearby ice floe.

“I was shocked,” Pitman said later. “It looked like they were trying to protect the seal.”

Perhaps they were. Humpbacks organize to protect their own calves, of course, but they also protect other species — indeed, this happened in nearly 90 percent of attacks where the killer whales’ prey could be identified.

Is this evidence of a moral sense among the whales? Donald Broom, an emeritus professor of animal welfare at Cambridge University, thinks so — if right means meeting individuals’ basic needs and maintaining their rights, and wrong means impeding these things, then there’s considerable evidence for a sense of morality among nonhumans. In The Cultural Lives of Whales and Dolphins, Hal Whitehead and Luke Rendell write, “When whales and dolphins go out of their way to help other creatures with their needs that looks pretty moral, at least on the surface.”

La Ola, or the “Mexican wave,” seems like the ultimate in spontaneous behavior, but biological physicist Illés Farkas of Eötvös University found that stadium waves can be studied quite effectively using the methods of statistical physics. Examining videos of waves in stadia holding more than 50,000 people, Farkas and his colleagues found that a crowd behaves like an excitable medium — the first group to stand acts as a “perturbation,” and in less than a second the wave begins, dying out on one side and continuing on the other (3 out of 4 Mexican waves travel clockwise around the stadium). And the speed of waves is surprisingly consistent — 22±3 seats per second, with an average width of 15 seats.

Farkas defined three states that a spectator can take: inactive (sitting, ready to stand), active (standing up), and refractory (sitting again afterward). Shizuoka University mechanical engineer Takashi Nagatani found that the local behavior of the spectators can be interpreted in terms of a chemically excitable medium with the following reaction set:

Passive + Activated ↦ 2 · Activated
Activated ↦ Refractory
Refractory ↦ Passive

Farkas wrote, “For a physicist, the interesting specific feature of this spectacular phenomenon is that it represents perhaps the simplest spontaneous and reproducible behaviour of a huge crowd with a surprisingly high degree of coherence and level of cooperation. In addition, La Ola raises the exciting question of the ways by which a crowd can be stimulated to execute a particular pattern of behaviour.”

(From Andrew Adamatzky, Dynamics of Crowd-Minds, 2005.)

A Friedman number, named after Stetson University mathematician Erich Friedman, is a number that can be calculated using its own digits, such as 736 = 3⁶ + 7 or 3281 = (3⁸ + 1) / 2.

A “nice” Friedman number is one in which the digits are used in order, such as 3685 = (3⁶ + 8) × 5 or 3972 = 3 + (9 × 7)².

Might this be done in other number systems? In a sense all Roman numerals are automatically Friedman numbers, but there are some interesting nontrivial examples as well:

XVIII = IV × II + X

LXXXIII = IX^X×X/L + II

And it turns out that “nice” examples are possible here too, in which a number’s letters are used in order:

LXXVI = L / X × XV + I

LXXXIV = LX / X × XIV

Is there more? Friedman has begun looking for examples in Mayan numerals — see his website.

Edric Cane came up with a simple way to establish any row in Pascal’s triangle, creating a simple sequence of fractions that, when multiplied successively, will produce the numbers in any desired row. Here’s an example for Row 7, giving the coefficients for (a + b)⁷ = a⁷ + 7a⁶b + 21a⁵b² + 35a⁴b³ + 35a³b⁴ + 21a²b⁵ + 7ab⁶ + b⁷:

Another example, for Row 10:

The same can be done for any desired row.

(Thanks, Alex.)

Mentioned in James Tanton’s Mathematics Galore!:

In 1740 the French mathematician Philippe Naudé sent a letter to Leonhard Euler asking how many ways a positive integer could be written as a sum of distinct positive integers (regardless of their order). In considering the problem Euler found something remarkable.

Let D(n) be the number of ways to write n as a sum of distinct positive integers. So, for example, D(6) is 4 because there are four ways to do this for 6: 6, 5 + 1, 4 + 2, and 3 + 2 + 1.

And let O(n) be the number of ways to write n as a sum of odd integers. So O(6) is 4 because 6 can be written as 5 + 1, 3 + 3, 3 + 1 + 1 + 1, or 1 + 1 + 1 + 1 + 1 + 1.

Euler showed that O(N) always equals D(N).