Mixed Emotions

A brainteaser by S. Ageyev, from the November-December 1991 issue of Quantum:

ageyev problem

Suppose that we change the signs of 50 of these numbers such that exactly half the numbers in each row and each column get a minus sign. Prove that the sum of all the numbers in the resulting table is zero.

Click for Answer


“The Mathematician in Love,” by Scottish mechanical engineer William Rankine (1820-1872):

A mathematician fell madly in love
With a lady, young, handsome, and charming:
By angles and ratios harmonic he strove
Her curves and proportions all faultless to prove
As he scrawled hieroglyphics alarming.

He measured with care, from the ends of a base,
The arcs which her features subtended:
Then he framed transcendental equations, to trace
The flowing outlines of her figure and face,
And thought the result very splendid.

He studied (since music has charms for the fair)
The theory of fiddles and whistles,–
Then composed, by acoustic equations, an air,
Which, when ’twas performed, made the lady’s long hair
Stand on end, like a porcupine’s bristles.

The lady loved dancing:–he therefore applied,
To the polka and waltz, an equation;
But when to rotate on his axis he tried,
His centre of gravity swayed to one side,
And he fell, by the earth’s gravitation.

No doubts of the fate of his suit made him pause,
For he proved, to his own satisfaction,
That the fair one returned his affection;–“because,
“As every one knows, by mechanical laws,
“Re-action is equal to action.”

“Let x denote beauty,–y, manners well-bred,–
z, Fortune,–(this last is essential),–
“Let L stand for love”–our philosopher said,–
“Then L is a function of x, y, and z,
“Of the kind which is known as potential.”

“Now integrate L with respect to d t,
“(t standing for time and persuasion);
“Then, between proper limits, ’tis easy to see,
“The definite integral Marriage must be:–
“(A very concise demonstration).”

Said he–“If the wandering course of the moon
“By Algebra can be predicted,
“The female affections must yield to it soon”–
–But the lady ran off with a dashing dragoon,
And left him amazed and afflicted.

The Mindset List

In 1998, Tom McBride and Ron Nief of Wisconsin’s Beloit College began compiling lists of what had “always” or “never” been true in the lives of each incoming class of students, to remind faculty to be mindful of the references they made in class.

For example, that first class, born in 1980, had been 11 years old when the Soviet Union broke up and did not remember the Cold War. They had never had a polio shot and never owned a record player. Their popcorn had always been cooked in a microwave, and they’d always had cable television. Here are some details of the worldview of the class of 2022:

  • Outer space has never been without human habitation.
  • They will never fly TWA, Swissair, or Sabena airlines.
  • The Prius has always been on the road in the U.S.
  • They never used a spit bowl in a dentist’s office.
  • “You’ve got mail” would sound as ancient to them as “number, please” would have sounded to their parents.
  • Mass market books have always been available exclusively as Ebooks.
  • There have always been more than a billion people in India.
  • Films have always been distributed on the Internet.
  • The detachable computer mouse is almost extinct.
  • The Mir space station has always been at the bottom of the South Pacific.

Other recent lists are here.

Tursiops Economicus


In the 1970s, dolphin trainer Jim Mullen sought to encourage the dolphins at Marine World in Redwood City, California, to tidy up their pool at the end of the day. Each dolphin received a reward of fish for each piece of litter it brought to him.

“It worked very well,” Mullen told psychologist Diana Reiss. “The pool was kept neat and clean, and the dolphins seemed to enjoy the game.”

One day in the summer of 1978, a dolphin named Spock seemed unusually diligent, bringing one piece after another of brown paper to Mullen and receiving a reward each time. Eventually Mullen grew suspicious and asked an assistant to go below and look through the pool windows.

“It turned out that there was a brown paper bag lodged behind an inlet pipe,” Mullen said. “Spock went to the paper bag, tore a piece off, and brought it to me. I then gave Spock a fish, as per our arrangement, and back he went. The second time my assistant saw Spock go to the paper bag, Spock pulled at it to remove a piece, but the whole bag came out. Spock promptly shoved the bag back into place, tore a small piece off, and brought it to me. He knew what he was doing, I’m sure. He completely had me.”

Spock hadn’t been trained to tear debris to pieces, and in doing so he was certainly maximizing his reward, Reiss writes. “And when he pushed the bag back behind the pipe when it came out in one piece, that certainly had the ring of deliberate action. Whether you can call it deliberate deception is a tough call.”

(Diana Reiss, The Dolphin in the Mirror, 2011.)

The Chameleon Vine

Image: Wikimedia Commons

Native to Chile and Argentina, Boquila trifoliolata has a remarkable ability: Once it’s wrapped its vines around a host plant, it can alter its leaves to mimic those of the host, a phenomenon called mimetic polymorphism.

“It modifies its size, shape, color, orientation, and even the pattern of its veins in such a way that it fuses perfectly with the foliage of the tree that bears it,” writes botanist Francis Hallé in his 2018 Atlas of Poetic Botany. “If, in the course of growing, it changes its support, the same stalk can even display leaves that are completely different, corresponding to the new tree — even if these leaves are much bigger.”

This helps it to avoid predators. If the plant grew along the forest floor it would be eaten by weevils, snails, and leaf beetles, but these tend to leave it alone when it disguises itself with “tree leaves.” But how it accomplishes the mimicry remains unclear.


When Louis Philippe was deposed, why did he lose less than any of his subjects?

Because, while he lost only a crown, they lost a sovereign.

— Edith Bertha Ordway, The Handbook of Conundrums, 1915

“Extra Magic” Realized

sallows toroidal square

From Lee Sallows:

The drawing at left above shows an unusual type of 3×3 geomagic square, being one in which the set of four pieces occupying each of the square’s nine 2×2 subsquares can be assembled so as to tile a 4×8 rectangle. The full set of subsquares become more apparent when it is understood that the square is to be viewed as if drawn on a torus, in which case its left-hand and right-hand edges will coincide, as will its upper and lower edges. In an earlier attempt at producing such a square several of the the pieces used were disjoint, or broken into separated fragments. Here, however, the pieces used are nine intact octominoes.

(Thanks, Lee!)

Going Home

Until 2006, a British ambassador leaving his post would write a valedictory despatch to be circulated among select readers in the British government. These “parting shots” tended to be appallingly frank, combining the diplomat’s real feelings about the nation he was leaving with his often wounded resentment at the indifference of his own government:

  • Argentina: “All I knew of Argentines before coming here was that they were generally disliked by all other Latin Americans as unduly pretentious, snobbish upstarts. …. The per capita outlay on deodorants in the Argentine is the highest in the world.”
  • Finland: “It could plausibly be argued that it is a misfortune for anybody but a Finn to spend three years in Finland, as I have just done. … Finland is flat, freezing, and far from the pulsating centres of European life.”
  • Uruguay: “After living now for over two years among people who call themselves Orientals and who seem to have inherited nearly all the faults and none of the virtues of Spain (though they have many minor virtues of their own), I look forward to returning to what I conceive to be, at least by contrast, the speed and efficiency of my own country.”
  • Saudi Arabia: “It is a great tragedy that, with all the world’s needs, Providence should have concentrated so much of a vital resource and so much wealth in the hands of people who need it so little and are so socially irresponsible about the use of it.”
  • Thailand: “Decayed garbage left for months on the side of the roads; stagnant canals that serve both as cesspools and as the dumping ground for dead dogs; buses and lorries that belch uncontrolled clouds of diesel fumes; scarcely a pavement without potholes and open manholes to break the legs of the unwary; bag-snatchers in every block; assault and violence a way of life; prostitution and every form of natural and unnatural vice on a scale astonishing even in Asia; a city of 4 million with only one park, and that littered with refuse and infested by thieves; unplanned hideous ribbon development; no proper drainage, so that in the rainy season large areas of the city remain flooded for weeks on end; and the whole set in a flat mournful plain without even a hillock in sight for 100 miles in any direction: this is Bangkok, the vaunted Venice of the East.”

Matthew Parris, who published a whole collection of these in 2010, explains: “Beyond retirement there can be no reprisals.”

The Bottom Line


In his 2008 book 100 Essential Things You Didn’t Know You Didn’t Know, cosmologist John D. Barrow considers how long a straight line a typical HB pencil could draw before the lead was exhausted.

A soft 2B pencil draws a line about 20 nanometers thick, and the diameter of a carbon atom is 0.14 nanometers, so a pencil line is only about 143 atoms thick. The pencil lead has a radius of about a millimeter, so its area is about π square millimeters. If the pencil is 15 centimeters long, then it contains 150π cubic millimeters of graphite.

Putting this together, if we draw a line 20 nanometers thick and 2 millimeters wide, then the pencil contains enough graphite to continue for the surprising distance L = 150π / 4 × 10-7 millimeters = 1,178 kilometers. “But I haven’t tested this prediction!”

(Thanks, Larry.)

05/21/2022 RELATED: How much of a pencil’s lead is wasted in the sharpening?

(Thanks, Chris.)

Birds of a Feather

A problem from the February 2006 issue of Crux Mathematicorum:

Prove that if 10a + b is a multiple of 7 then a – 2b must be a multiple of 7 as well.

Click for Answer