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).

From Martin Gardner: Shuffle a deck of cards and start dealing them face up one at a time into a pile, counting down from 10 as you go — that is, say “10” as you deal the first card, “9” as you deal the second, and so on. As soon as you say a number that matches the value of the card you’re dealing, stop dealing and start a new pile. Face cards count as 10. If you count down to 1 without finding a match, “kill” the pile by putting a card face down on top of it.

Repeat this procedure until you’ve dealt four piles. (If all four piles are “killed,” which is very unlikely, shuffle the whole deck and start over.) When the four piles are finished, add the values of the cards at the top of each “living” pile. Call that sum k. Now deal k cards from the remainder of the deck, and then count the cards that remain.

There will always be eight.

(Martin Gardner, “Curious Counts,” Math Horizons 10:3 [February 2003], 20-22.)

Given the premises No fruit-picker is a sailor and All Ruritanians are fruit-pickers, it’s fairly easy to deduce that No Ruritanian is a sailor. But what logical conclusion can be drawn from these premises?

All members of the cabinet are thieves.

No composer is a member of the cabinet.

Many people decide that no conclusion can be drawn. “Almost without exception that is the answer you will get, after some serious reflection, from intelligent people,” notes Massimo Piattelli-Palmarini in Inevitable Illusions.

But there is one: Some thieves are not composers (or There are thieves who are not composers). It’s logically impossible for the premises to be true and for this conclusion not to be true, and yet most people find the conclusion difficult to see.

Why? Princeton psychologist Philip Johnson-Laird thinks it’s because of the number and complexity of “mental models” that we have to build to elaborate our reasoning about such syllogisms. Unlike the case of the Ruritanian sailors, “The case of the cabinet ministers and composers requires … three distinct, and mentally separate, arrays of obligatory couplings: cabinet ministers and thieves, composers and cabinet ministers, and thieves and composers,” Piattelli-Palmarini explains.

“The result for all of us is a prohibitive difficulty in ‘seeing’ that there are necessarily false couplings between thieves and composers.”

01/04/2018 A number of readers have pointed out the existential fallacy in Johnson-Laird’s syllogism. Piattelli-Palmarini notes, “[O]ne has to avoid the situation in which these sets are empty. Rephrasing the premises as ‘all the ministers’ and ‘all the composers’ may aptly reinforce in the subjects the assumption that there are ministers and that there are composers. As the distinguished logician George Boolos of MIT has put it, the sentence ‘All deserters will be shot’ can be true also if there are, in fact, no deserters. In these cognitive tests, one wants to avoid such limited cases. For a further and accurate account of this problem, the reader is referred to Philip Johnson-Laird’s Mental Models, Chapter 6, and to his exchange with George Boolos in the journal Cognition in 1984.” (That’s Cognition 17:2, 181-182; Johnson-Laird’s reply is in 17:2, 183-184.)