“The light in the world comes principally from two sources, — the sun, and the student’s lamp.” — Christian Nestell Bovee, Intuitions and Summaries of Thought, 1862



Create a strip of 19 triangles like the one above (printable version here) and fold the left portion back successively at each of the northeast-pointing lines to produce a spiral:

Fold this spiral backward along line ab:

Then fold the resulting figure backward at cd. You should be left with one blank triangular tab that can be folded backward and pasted to another blank panel on the opposite side. The resulting hexagon should have six 1s on one side and six 2s on the other.

With some adroit pinching this hexagon produces some marvelous effects. Fold down two adjacent triangles so that they meet, and then press in the opposite corner to join them. Now the top of the figure can be prised open and folded down to produce a new hexagon — this one with 1s on one face and a surprising blank on the second. What has become of the 2s?

Exploring the properties of this “hexahexaflexagon” offers an intuitive lesson in geometric group theory:

When Martin Gardner wrote about these bemusing creatures in his first column for Scientific American in 1956, he received two letters. The first was from Neil Uptegrove of Allen B. Du Mont Laboratories in Clifton, N.J.:


I was quite taken with the article entitled ‘Flexagons’ in your December issue. It took us only six or seven hours to paste the hexahexaflexagon together in the proper configuration. Since then it has been a source of continuing wonder.

But we have a problem. This morning one of our fellows was sitting flexing the hexahexaflexagon idly when the tip of his necktie became caught in one of the folds. With each successive flex, more of his tie vanished into the flexagon. With the sixth flexing he disappeared entirely.

We have been flexing the thing madly, and can find no trace of him, but we have located a sixteenth configuration of the hexahexaflexagon.

Here is our question: Does his widow draw workmen’s compensation for the duration of his absence, or can we have him declared legally dead immediately? We await your advice.

The second was from Robert M. Hill of The Royal College of Science and Technology in Glasgow, Scotland:


The letter in the March issue of your magazine complaining of the disappearance of a fellow from the Allen B. Du Mont Laboratories ‘down’ a hexahexaflexagon, has solved a mystery for us.

One day, while idly flexing our latest hexahexaflexagon, we were confounded to find that it was producing a strip of multicolored material. Further flexing of the hexahexaflexagon finally disgorged a gum-chewing stranger.

Unfortunately he was in a weak state and, owing to an apparent loss of memory, unable to give any account of how he came to be with us. His health has now been restored on our national diet of porridge, haggis and whisky, and he has become quite a pet around the department, answering to the name of Eccles.

Our problem is, should we now return him and, if so, by what method? Unfortunately Eccles now cringes at the very sight of a hexahexaflexagon and absolutely refuses to ‘flex.’

Flash Mob

Pipe plot - 1877 - George Henry Boughton

When Wilhelm Kieft tried to outlaw smoking in New Amsterdam in the 1630s, he brought on a unique protest. Washington Irving writes:

A mob of factious citizens had … the hardihood to assemble before the governor’s house, where, setting themselves resolutely down, like a besieging army before a fortress, they one and all fell to smoking with a determined perseverance, that seemed as though it were their intention to smoke him into terms. The testy William issued out of his mansion like a wrathful spider, and demanded to know the cause of this seditious assemblage, and this lawless fumigation; to which these sturdy rioters made no other reply, than to loll back phlegmatically in their seats, and puff away with redoubled fury; whereby they raised such a murky cloud, that the governor was fain to take refuge in the interior of his castle.

Wilhelm finally gave in — people could smoke, he said, but they had to give up long pipes. “Thus ended this alarming insurrection, which was long known by the name of the pipe plot, and which, it has been somewhat quaintly observed, did end, like most other plots, seditions, and conspiracies, in mere smoke.”

(Thanks, Dan.)

Black and White

yarosh chess puzzle

By Alexander Yarosh. The position above was reached in a legal game, except that one piece has been knocked off the board. What was it?

Click for Answer

Saving Face

“The Joker,” a picture-preserving geomagic square by Lee Sallows. The 16 pieces can be assembled in varying groups of 4 to produce the same picture in 16 different ways, without rotation or reflection.

The outline need not be a joker — it can take almost any shape.

In a Word


n. “Name for a devil said to collect fragments of words dropped, skipped, or mumbled in the recitation of divine service, and to carry them to hell, to be registered against the offender.” [OED]

Low Profile


In 2008 L.A. Innes of Jamestown, Saint Helena, auctioned a collection of images taken during the Boer War. This one shows a prisoner standing next to a tortoise on the island. The tortoise was mature at the time of the photograph, which was taken in 1900, and investigators were surprised to find that he’s still alive — “Jonathan” lives on the grounds of the governor’s residence, blind in one eye but still active and mating with other tortoises.

If he was 70 at the time Innes’ photograph was taken, then he’s 184 today — the oldest living reptile on earth.

Small World


I don’t think this was ever built — in 1904 engineer Hiram Stevens Maxim designed an amusement with a rotating parabolic floor “for producing illusionary effects”:

With such a contrivance when persons enter the hollow sphere they will not be able to tell whether it is revolving or standing still and by reason of the parabolic floor, persons near the outer edge would, to the persons standing near the centre, appear to be walking with their heads directed inward. When the sphere revolves some curious phenomena will be obtained in walking outward and inward on such a floor, and the throwing of a ball from the centre outward and vice versa will move in an unexpected direction that will be very puzzling to the people in the sphere.

Fig. 3, below, shows the perspective from the edge of the floor as it rotates. If mirrors were positioned overhead, as in Fig. 4, “people could then be made to appear to be walking all over the inside of the sphere with their heads pointing inward and their feet pointing outward.”


The Just World Phenomenon

In 1966 University of Kentucky social psychologist Melvin Lerner asked 72 undergraduate women to observe a peer working on a learning task. When the learner made an error she appeared to receive a painful electric shock. In describing her suffering, the observers tended to reject and devalue her when they thought they would continue to see her suffer in a later session.

Lerner suggested that we come to terms with the suffering we see around us by deciding that the world is just — that those who are unfortunate somehow deserve their fate, and thus that we can avoid such a fate ourselves. This is reflected in figures of speech such as “You reap what you sow” and “He got what was coming to him.”

“If people did not believe that they could get what they want and avoid what they abhor by performing certain appropriate acts, they would be virtually incapacitated,” Lerner wrote. “If this is true, then the person who sees suffering or misfortune will be motivated to believe that the unfortunate victim in some sense merited his fate.”

(Melvin J. Lerner and Carolyn H. Simmons, “Observer’s reaction to the ‘innocent victim’: Compassion or rejection?”, Journal of Personality and Social Psychology 4(2) [August 1966], 203-210.)

Seeing Double


In 1944, as the Allies were preparing to invade France, British Intelligence sought a way to confuse the Germans as to their plans. They hired Meyrick Clifton James (right), an Australian-born lieutenant in the Army Pay Corps who bore a striking resemblance to Field Marshal Bernard Montgomery, who would be commanding the Allied ground troops during the invasion. David Niven, then a colonel in the Army Film Unit, invited James to London under cover as a journalist, and James set about studying the general’s speech patterns and mannerisms. Then he was conspicuously dispatched, as Monty, to Gibraltar and then to Algiers, watched by avid German spies.

It seemed to work. The plot went through “from start to finish without a hitch,” MI5 reported, “and we knew that the main feature of its story had reached the Germans.” The real Monty led the successful landings at Normandy while James recovered from the ordeal in a safe house in Cairo. “He was under terrible pressure and strain,” reported the wife of an intelligence officer detailed to look after him. “Coming out of that part was very difficult for him.” But he had some consolation: Under army rules, he would receive the equivalent of a general’s pay for every day he had impersonated Monty.

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