The Angel Problem
Image: Wikimedia Commons

An angel stands on an infinite chessboard. On each turn she can move at most 3 king’s moves from her current position. Play then passes to a devil, who can eat any square on the board. The angel can’t land on an eaten square, but she can fly over it, as angels have wings. (In the diagram above, the angel starts at the origin of the grid and, since she’s limited to 3 king’s moves, can’t pass beyond the blue dotted boundary on her next turn.)

The devil wins if he can strand the angel by surrounding her with a “moat” 3 squares wide. The angel wins if she can continue to move forever. Who will succeed?

John Conway, who posed this question in 1982, offered $100 for a winning strategy for an angel of sufficiently high “power” (3 moves may not be enough; in fact a 1-power angel, an actual chess king, will lose). He also offered $1000 for a strategy that will enable a devil to win against an angel of any power.

It’s not immediately clear what strategy can save the angel. If she simply flees from nearby eaten squares, the devil can build a giant horseshoe and drive her into it. If she sprints in a single direction, the devil can build an impenetrable wall to stop her.

In fact it wasn’t until 2006 that András Máthé and Oddvar Kloster both showed that the angel has a winning strategy. In some variants, in higher dimensions, it’s still not certain she can survive.

(John H. Conway, “The Angel Problem,” in Richard J. Nowakowski, ed., Games of No Chance, 1996.)


  • Conceptual artist Joseph Beuys accepted responsibility for any snow that fell in Düsseldorf February 15-20, 1969.
  • Any three of the numbers {1, 22, 41, 58} add up to a perfect square.
  • Nebraska is triply landlocked — a resident must cross three states to reach an ocean, gulf, or bay.
  • The only temperature represented as a prime number in both Celsius and Fahrenheit is 5°C (41°F).
  • “A person reveals his character by nothing so clearly as the joke he resents.” — Georg Christoph Lichtenberg

“I was tossing around the names of various wars in which both the opponents appear: Spanish-American, Franco-Prussian, Sino-English, Russo-Japanese, Arab-Israeli, Judeo-Roman, Anglo-Norman, and Greco-Roman. Is it a quirk of historians or merely a coincidence that the opponent named first was always the loser? It would appear that a country about to embark on war would do well to see that the war is named before the fighting starts, with the enemy named first!” — David L. Silverman

Podcast Episode 162: John Muir and Stickeen

One stormy morning in 1880, naturalist John Muir set out to explore a glacier in Alaska’s Taylor Bay, accompanied by an adventurous little dog that had joined his expedition. In this week’s episode of the Futility Closet podcast we’ll describe the harrowing predicament that the two faced on the ice, which became the basis of one of Muir’s most beloved stories.

We’ll also marvel at some phonetic actors and puzzle over a season for vasectomies.


In 1904 a 12-year-old J.R.R. Tolkien sent this rebus to a family friend.

In 1856 Preston Brooks beat Charles Sumner with a gold-headed cane on the floor of the U.S. Senate.

Sources for our feature on John Muir and Stickeen:

John Muir, Stickeen, 1909.

Ronald H. Limbaugh, John Muir’s “Stickeen” and the Lessons of Nature, 1996.

Kim Heacox, John Muir and the Ice That Started a Fire, 2014.

Ronald H. Limbaugh, “Stickeen and the Moral Education of John Muir,” Environmental History Review 15:1 (Spring 1991), 25-45.

Hal Crimmel, “No Place for ‘Little Children and Tender, Pulpy People’: John Muir in Alaska,” Pacific Northwest Quarterly 92:4 (Fall 2001), 171-180.

Stefan Beck, “The Outdoor Kid,” New Criterion 33:4 (December 2014), 1-6.

Edward Hoagland, “John Muir’s Alaskan Rhapsody,” American Scholar 71:2 (Spring 2002), 101-105.

Ronald H. Limbaugh, “John Muir and Modern Environmental Education,” California History 71:2 (Summer 1992), 170-177.

Oxford Dictionary of National Biography, “John Muir” (accessed July 2, 2017).

“John Muir: Naturalist,” Journal of Education 81:6 (Feb. 11, 1915), 146.

William Frederic Badè, “John Muir,” Science 41:1053 (March 5, 1915), 353-354.

Charles R. Van Hise, “John Muir,” Science 45:1153 (Feb. 2, 1917), 103-109.

Listener mail:

Delta Spirit, “Ballad of Vitaly”:

Wikipedia, Aftermath (2017 Film)” (accessed July 14, 2017).

Wikipedia, “Überlingen Mid-Air Collision” (accessed July 14, 2017).

Anthony Breznican, “‘The Princess Bride’: 10 Inconceivable Facts From Director Rob Reiner,” Entertainment Weekly, Aug. 16, 2013.

Wikipedia, “Charlotte Kate Fox” (accessed July 14, 2017).

Wikipedia, Incubus (1966 film)” (accessed July 14, 2017).

Wikipedia, “Esperanto” (accessed July 14, 2017).

Toño del Barrio, “Esperanto and Cinema” (accessed July 14, 2017).

Wikipedia, “Phonetical Singing” (accessed July 14, 2017).

Wikipedia, “Deliver Us (The Prince of Egypt)” (accessed July 14, 2017).

This week’s lateral thinking puzzle was inspired by an item in Dan Lewis’ Now I Know enewsletter. (Warning: This link spoils the puzzle.)

You can listen using the player above, download this episode directly, or subscribe on iTunes or Google Play Music or via the RSS feed at

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation on the Support Us page of the Futility Closet website.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at Thanks for listening!

A Little Story

IBM nanophysicists have made a stop-motion movie using individual atoms — carbon monoxide molecules arranged on a copper substrate and then magnified 100 million times using a scanning tunneling microscope. The molecules remain stationary because they form a bond with the substrate at this extremely low temperature (-268.15° C); each CO molecule stands “on end” so that only one atom is visible.

The result, “A Boy and His Atom,” holds the Guinness world record for the world’s smallest stop-motion film.

07/16/2017 UPDATE: The day I posted this, news broke that researchers have encoded five frames of an 1870s short film into bacterial DNA. (Thanks, Sharon.)

Monkey See
Image: Wikimedia Commons

In 1972 biologists Colin Tayler and Graham Saayman were observing a group of Indo-Pacific bottlenose dolphins in a South African aquarium. One of them, a 6-month-old calf named Dolly, began to seek their attention by pressing feathers, stones, seaweed, and fish skins against the glass of the viewing chamber. If they ignored her she swam off and returned with a different object.

At the end of one observation session, one of the investigators blew a cloud of cigarette smoke against the glass as Dolly was looking in. “The observer was astonished when the animal immediately swam off to its mother, returned and released a mouthful of milk which engulfed her head, giving much the same effect as had the cigarette smoke,” the biologists reported. “Dolly subsequently used this behaviour as a regular device to attract attention.”

“Dolly didn’t ‘copy’ (she wasn’t really smoking) or imitate with intent to achieve the same purpose,” argues ecologist Carl Safina in Beyond Words: What Animals Think and Feel. “Somehow Dolly came up with the idea of using milk to represent smoke. Using one thing to represent something else isn’t just mimicking. It is art.”

(C.K. Tayler and G.S. Saayman, “Imitative Behaviour by Indian Ocean Bottlenose Dolphins [Tursiops aduncus] in Captivity,” Behaviour 44:3 [1973], 286-298.)

Noted in Passing

In Visual Thinking in Mathematics, M. Giaquinto writes, “Calculus grew out of attempts to deal with quantitative physical problems which could not be solved by means of geometry and arithmetic alone. Many of these problems concern situations which are easy to visualize. In fact visual representations are so useful that most books on calculus are peppered with diagrams.” But there’s an intriguing footnote: “Moshé Machover brought to my attention a notable exception: Landau (1934). It has no diagram, and no geometrical application.”

That’s Differential and Integral Calculus, by Edmund Landau, a professor of mathematics at Gottingen University. Machover is right — the 366-page volume contains not a single diagram. Landau writes, “I have not included any geometric applications in this text. The reason therefor is not that I am not a geometer; I am familiar, to be sure, with the geometry involved. But the exposition of the axioms and of the elements of geometry — I know them well and like to give courses on them — requires a separate volume which would have to precede the present one. In my lecture courses on the calculus, the geometric applications do, of course, make up a considerable portion of the material that is covered. But I do not wish to wait any longer to make generally available an account, rigorous and complete in every particular, of that which I have considered in my courses to be the most suitable method of treating the differential and integral calculus.”

The book was quite successful — the first English edition appeared in 1950, and subsequent editions have continued right up through 2001.

His and Hers

russell illusion

Which of these faces is male, and which female? In fact both photos show the same androgynous face; the only difference is the amount of contrast in the image. But most people see the face on the left as female and the one on the right as male.

Gettysburg College psychologist Richard Russell says, “Though people are not consciously aware of the sex difference in contrast, they unconsciously use contrast as a cue to tell what sex a face is. We also use the amount of contrast in a face to judge how masculine or feminine the face is, which is related to how attractive we think it is.”

Cosmetics may serve to make a female face more attractive by heightening this contrast. “Cosmetics are typically used in precisely the correct way to exaggerate this difference,” Russell says. “Making the eyes and lips darker without changing the surrounding skin increases the facial contrast. Femininity and attractiveness are highly correlated, so making a face more feminine also makes it more attractive.”

(Richard Russell, “A Sex Difference in Facial Pigmentation and Its Exaggeration by Cosmetics,” Perception 38:8 [August 2009], 1211-1219.)

Higher Magic

The digits 1-9 can be arranged into a 3 × 3 magic square in essentially one way (not counting rotations or reflections) — the so-called lo shu square:

4    3    8

9    5    1

2    7    6

As in any magic square, each row, column, and diagonal produces the same total. But surprisingly (to me), the sum of the row products also equals the sum of the column products:

4 × 3 × 8 + 9 × 5 × 1 + 2 × 7 × 6 = 96 + 45 + 84 = 225

4 × 9 × 2 + 3 × 5 × 7 + 8 × 1 × 6 = 72 + 105 + 48 = 225

Even more surprisingly, the same is true of the Fibonacci sequence, if we arrange its first nine terms into a square array in the same pattern:

 3    2   21

34    5    1

 1   13    8

3 × 2 × 21 + 34 × 5 × 1 + 1 × 13 × 8 = 126 + 170 + 104 = 400

3 × 34 × 1 + 2 × 5 × 13 + 21 × 1 × 8 = 102 + 130 + 168 = 400

It turns out that this is true of any second-order linear recursion. (The sums won’t always be squares, though.)

From Edward J. Barbeau’s Power Play, 1997.

A Twist in History,_Eindeloze_kronkel,_1953-1956.jpg
Image: Wikimedia Commons

Swiss artist Max Bill conceived the Möbius strip independently of August Möbius, who discovered it in 1858. Bill called his figure Eindeloze Kronkel (“Endless Ribbon”), after the symbol of infinity, ∞, and began to exhibit it in various sculptures in the 1930s. He recalled in a 1972 interview:

I was fascinated by a new discovery of mine, a loop with only one edge and one surface. I soon had a chance to make use of it myself. In the winter of 1935-36, I was assembling the Swiss contribution to the Milan Triennale, and there was able to set up three sculptures to characterize and accentuate the individuality of the three sections of the exhibit. One of these was the Endless Ribbon, which I thought I had invented myself. It was not long before someone congratulated me on my fresh and original reinterpretation of the Egyptian symbol of infinity and of the Möbius ribbon.

He pursued mathematical inspirations actively in his later work. He wrote, “The mystery enveloping all mathematical problems … [including] space that can stagger us by beginning on one side and ending in a completely changed aspect on the other, which somehow manages to remain that selfsame side … can yet be fraught with the greatest moment.”