Math Notes

121213882349 = (1212 + 1388 + 2349)3

128711132649 = (1287 + 1113 + 2649)3

162324571375 = (1623 + 2457 + 1375)3

171323771464 = (1713 + 2377 + 1464)3

368910352448 = (3689 + 1035 + 2448)3


Mike Keith found this amazing correspondence in 2004. The two 6×6 squares below contain 72 different entries from the periodic table of the elements:

mike keith chemical squares

The two squares are equal in three different ways:

  1. If you spell out the name of each element listed (hydrogen, beryllium, etc.), the square on the left is an anagram of the square on the right.
  2. The sum of the atomic numbers of the 36 elements on the left (2019) equals the sum of those on the right.
  3. If you replace each symbol with its alphabetic score (where A=1, B=2, etc.; e.g. Li = L + I = 12 + 9 = 21), then the sum of the scores on the left (737) equals that of those on the right.

Keith writes, “The next largest pair of triply-equal squares like this would be 7×7 in size, containing a total of 98 different elements, [and] it seems quite unlikely that 98 of them could be so arranged. If this is true then the 6×6 pair presented here is the largest possible (at least for now, until many more new chemical elements have been discovered and named).”

(Mike Keith, “A Magical Pair of 6×6 Chemical Squares,” Word Ways, February 2004.)

Sylver Coinage

This curious game was invented by Princeton mathematician John H. Conway. Two players take turns naming positive integers, but an integer is off limits if it’s the sum of nonnegative multiples of integers that have already been named. Once 1 is named, everything is off limits (because any positive integer is a sum of 1s), so that ends the game; the player who is forced to name 1 is the loser. An example gives the idea:

  • I start by naming 5. From now on neither of us can name 5, 10, 15, 20, …
  • You name 4. Now neither of us can name a number built of 5s and 4s, that is, 4, 5, 8, 9, 10, or any number greater than 11.
  • I name 11. This reduces the list of available numbers to 1, 2, 3, 6, and 7.
  • You name 6. Now we’re down to 1, 2, 3, and 7.
  • I name 7. Only 1, 2, and 3 are still available to name.

Now we’ll use our next turns to name 2 and 3, and you’ll be forced to name 1, losing the game.

Though the game is very easy to understand, it’s still full of mysteries. For example, R.L. Hutchings has shown that playing a prime number as the first move guarantees that a winning strategy exists, but no one has figured out how to find the strategy. And no one knows whether there are any winning opening moves at all that aren’t prime.

More on the Sylver Coinage page.


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.


Rudyard Kipling hid messages in his illustrations for the Just So Stories.

In the early 1900s, Danes bred pigs colored to resemble the Danish flag.

Sources for our feature on Augustine Courtauld:

Nicholas Wollaston, The Man on the Ice Cap, 1980.

Mollie Butler, August and Rab, 1987.

“Augustine Courtauld,” Encyclopedia Arctica (accessed July 23, 2017).

“Augustine Courtauld,” Oxford Dictionary of National Biography (accessed July 23, 2017).

“The British Arctic Air Route Expedition,” Geographical Journal 76:1 (July 1930), 67-68.

“British Air Route to the Arctic Regions,” Science, New Series, 72:1857 (Aug. 1, 1930), 108-109.

“Swedish Flier Ready to Hop for Greenland to Rescue Courtauld, Young British Explorer,” New York Times, April 27, 1931, 4.

Svend Carstensen, “Ahrenberg to Start Rescue Flight Today,” New York Times, April 29, 1931, 12.

Svend Carstensen, “Ahrenberg on Way to Save Courtauld, Lost in Greenland,” New York Times, April 30, 1931, 1.

“Rescuers Race to Locate Lost Arctic Explorer,” China Press, May 2, 1931, 13.

E. Lemon, “Plane in Greenland to Hunt Courtauld,” New York Times, May 3, 1931, 2.

Percy Lemon, “Ahrenberg Ready to Fly to Ice Cap,” New York Times, May 5, 1931, 6.

“Courtauld Hunted by Sea, Air And Land: Area of Great Arctic Search,” New York Times, May 8, 1931, 12.

“Courtauld Rescued,” Los Angeles Times, May 8, 1931, 3.

Percy Lemon, “Courtauld Is Found Safe on the Greenland Ice Cap,” New York Times, May 8, 1931, 1.

Albin Ahrenberg, “Ahrenberg to Guide Courtauld To Camp,” New York Times, May 9, 1931, 1.

Percy Lemon, “Courtauld Back Safely on Greenland Coast,” New York Times, May 12, 1931, 1.

H.G. Watkins, “Courtauld Search a Surprise to Him,” New York Times, May 14, 1931, 12.

“Courtauld Buried in Igloo 2 Months,” Associated Press, May 15, 1931.

“Arctic Burial Escape Told,” Los Angeles Times, May 15, 1931, 4.

“Courtauld Tells Story of Long Imprisonment,” China Press, May 15, 1931, 1.

“Rescued From Greenland’s Icy Cap,” Sphere 125:1634 (May 16, 1931), 278.

“Courtauld to Sail Home on First Ship,” New York Times, May 17, 1931, 2.

T.J.C. Martyn, “Greenland Is Still a Scientific Puzzle,” New York Times, May 24, 1931, 4.

Augustine Courtauld, “Courtauld’s Story of the Five Months He Spent on Ice Cap,” New York Times, May 29, 1931, 1.

“The Ice-Cap Hero,” New York Times, May 30, 1931, 8.

“The British Arctic Air Route Expedition,” Geographical Journal 77:6 (June 1931), 551-554.

“From the Four Winds: Mr. Courtauld’s Arctic Vigil,” China Herald, June 30, 1931, 459.

“The British Arctic Air Route Expedition,” Geographical Journal 78:3 (September 1931), 291.

F.S. Chapman, “Watkins and Aides Held in No Danger,” New York Times, Sept. 19, 1931, 17.

“Explorers Return From Greenland,” New York Times, Nov. 14, 1931, 8.

William Goodenough, Augustine Courtauld, Lauge Koch, J.M. Wordie, and H.R. Mill, “The British Arctic Air Route Expedition: Discussion,” Geographical Journal 79:6 (June 1932), 497-501.

Percy Cox, Helge Larsen, Augustine Courtauld, M.A. Spender and J.M. Wordie, “A Journey in Rasmussen Land: Discussion,” Geographical Journal 88:3 (September 1936), 208-215.

Henry Balfour, E.C. Fountaine, W.A. Deer, Augustine Courtauld, L.R. Wager, and Ebbe Munck, “The Kangerdlugssuak Region of East Greenland: Discussion,” Geographical Journal 90:5 (November 1937), 422-425.

“Augustine Courtauld Dies at 54: Explored Greenland in Thirties,” New York Times, March 4, 1959, 31.

L.R. Wager, “Mr. Augustine Courtauld,” Nature 183:4666 (April 4, 1959).

Quintin Riley, “Obituary: Augustine Courtauld 1904-1959,” Geographical Journal 125:2 (June 1959), 286-287.

Ronald Porter, “Lady Butler of Saffron Walden,'” Independent, April 1, 2009.

Listener mail:

Matthew J. Kinservik, Sex, Scandal, and Celebrity in Late Eighteenth-Century England, 2007.

Chris Best, “Watch: Hungry Bear Opens Fridge, Rummages Through Home,”, July 6, 2017.

“NC Bear Opens SUV Door, Climbs Inside and Destroys It,”, July 8, 2017.

Mark Price, “NC’s Bears Are Now Opening Car Doors, Leading to Strange Driveway Encounters,” Charlotte Observer, July 9, 2017.

“Bear and the SUV,” Sylva Herald, June 21, 2017.

This week’s lateral thinking puzzle was devised by Sharon. Here are three corroborating links (warning — these spoil the puzzle).

You can listen using the player above, download this episode directly, or subscribe on Apple Podcasts 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 or browse our online store for Futility Closet merchandise.

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 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