Here’s an especially vivid example of the illusion created by Dick Termes’ six-point perspective.

If you can convince yourself that the front half of this sphere is transparent, and that the image is painted on the interior of the back half, you’ll find that you’re inside the cage, turning to your left, while the birds are outside the cage, looking in at you. (To get started, I find it helps to focus on an edge of the sphere, rather than the center.)

There are many more examples on Termes’ YouTube channel.

The Last Detail

A puzzle by R.P. Cross:

Find the last digit in the evaluation of  \sum_{n = 1}^{100}n!

Click for Answer

Podcast Episode 164: Vigil on the Ice

Image: Wikimedia Commons

In 1930, British explorer Augustine Courtauld volunteered to spend the winter alone on the Greenland ice cap, manning a remote weather station. As the snow gradually buried his hut and his supplies steadily dwindled, his relief party failed to arrive. In this week’s episode of the Futility Closet podcast we’ll follow Courtauld’s increasingly desperate vigil on the ice.

We’ll also retreat toward George III and puzzle over some unexpected evidence.

See full show notes …

A Good Start

opalka 1 million

In 1965, Polish artist Roman Opałka hung a 196 × 135 cm canvas in his Warsaw studio. In the top left corner he painted a tiny numeral 1, then a 2, and so on until he had filled the canvas with numbers. Then he put up a new canvas and continued where he had left off. He called these images “details”; all of them had the same size and the same title, 1965 / 1 – ∞.

He vowed to spend the rest of his life on the project. “All my work is a single thing,” he said, “the description from number one to infinity. A single thing, a single life. … The problem is that we are, and are about not to be.”

At the start he painted white numbers on a black background, but in 1972 he began gradually to lighten the black with each detail, saying that his goal was “to get up to the white on white and still be alive.” He expected that this would happen when he reached 7777777 … but at the time of his death, in 2011, he’d got only as far as 5607249.

Mens Agitat Molem


In 2010 Jeremy Wood walked around the campus of the University of Warwick with a GPS device to “draw” a map at 1:1 scale. Altogether he covered 238 miles in 17 days.

“He stayed in the Maths Houses on Gibbet Hill so the line through Tocil Wood to the Mead Gallery is exceptionally dark since it was walked so many times,” the university reports. “As he worked his way across the fields towards Kenilworth he began to ‘draw’ images associated with the University, from its crest, to a mortar board, to a globe in homage to the many ‘international’ centres that he encountered in his journeys. Reported to security several times for walking in ‘a suspicious manner’ around Claycroft and Lakeside residences, he soon disappeared from view, walking the countryside that surrounds the University but which is far removed from central campus.”

“I responded to the structure of each location and avoided walking along roads and paths when possible,” Wood writes. “Security was called on me twice on separate occasions and I lost count of how many times I happened to trigger an automatic sliding door.” More at his website and at GPS Drawing.

Somewhat related: Mathematician Jerry Farrell invented a two-player coin-pushing game played on a map of Butler University, his institution. Rebecca Wahl analyzed it in Barry Cipra’s Tribute to a Mathemagician (2005), and Aviezri Fraenkel of Israel’s Weizmann Institute of Science revisited it the following year (PDF).

Changing Views

Sculptor John V. Muntean writes, “Our scientific interpretation of nature often depends upon our point of view. Perspective matters.” His carving titled Riddle of the Sphinx combines three profiles in one object — a baby, an adult man, and an old man with a walking stick. With each 120 degrees of rotation, the carving’s shadow presents a different picture.

This is impressive enough when it’s carved in mahogany, but Muntean’s Knight Mermaid Pirate Ship, below, works the same trick using LEGOs.

See more of Muntean’s “magic angle sculptures” on his YouTube channel.

Math Notes

(15 + 25) + (17 + 27) = 2 (1 + 2)4

(15 + 25 + 35) + (17 + 27 + 37) = 2 (1 + 2 + 3)4

(15 + 25 + 35 + 45) + (17 + 27 + 37 + 47) = 2 (1 + 2 + 3 + 4)4

(15 + 25 + 35 + 45 + 55) + (17 + 27 + 37 + 47 + 57) = 2 (1 + 2 + 3 + 4 + 5)4

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