The Szilassi Polyhedron
Image: Wikimedia Commons

In the humble tetrahedron, each face shares an edge with each other face. Surprisingly, there’s only one other known polyhedron in which this is true — the Szilassi polyhedron, discovered in 1977 by Hungarian mathematician Lajos Szilassi:

If there’s a third such creature it would have 44 vertices and 66 edges, and no one knows whether such a shape could even be contrived. It remains an unsolved problem.

Square Deal

Arrange the first n2 odd numbers in a square (here n = 6):

 1  3  5  7  9 11
13 15 17 19 21 23
25 27 29 31 33 35
37 39 41 43 45 47
49 51 53 55 57 59
61 63 65 67 69 71

Now, no matter n, the sum of the first row is n2, the sum of either long diagonal is n3, and the sum of the whole array is n4.

(From Edward Barbeau’s Power Play, 1997.)


In 2006, screenwriter Gregory K. Pincus invited the readers of his blog to submit “Fibs,” poems of six lines whose syllable counts reflect the Fibonacci sequence:

Spiraling mixture:
Math plus poetry yields the Fib.

Predictably, this took off on Slashdot, where it spawned a thousand variations:

01 It
01 is
02 really
03 not taxing
05 to create a Fib,
08 but still they are interesting
13 sequences of numbers. We are familiar with
21 the ‘rabbit generation’ origins of the sequence, but it can also describe
34 the number of petals on a flower, or the number of curves on a sunflower head, on a pineapple, or even on a pinecone.

And from there it expanded around the world. “The success of this story was entirely because the poem was based on the Fibonacci sequence,” Slashdot founder Rob Malda told the Poetry Foundation. “Geeks love interesting number sequences, and that one is way up there. Generally speaking literature by itself isn’t our typical subject matter, but interesting use of math definitely is.”

“To my surprise (and joy), I continue to find new threads of Fibs popping up all around the Web,” wrote Pincus, who eventually parlayed the idea into a novel. “I’ve seen Fibs in over a dozen different languages, and I’d also note that today a cat left a post in the comments of The Fib, joining a priorly poetic dog, so I think it’s safe to say that Fibs travel well.”

Acquitting Oneself
Image: Wikimedia Commons

In Circularity, Ron Aharoni mentions a story by Raymond Smullyan. On a certain island there are two kinds of people, those who always lie and those who always tell the truth. One day an islander is arrested on suspicion of murder. At his trial he says, “The murderer is a liar.”

Smullyan argues that this piece of evidence alone should acquit him. If the man is honest, then what he says is true, the murderer is a liar, and since he himself is a truth-teller he cannot be the guilty party. On the other hand, if he’s a liar then his testimony is false, which means that the murderer is in fact not a liar, and once again he cannot be guilty. Either way, he proves his innocence by showing that the murderer and himself belong to two different tribes.

Aharoni adds, “The problem is that the man was found beside the corpse with a bloody knife in his hand and a wide smile on his face. He is obviously the murderer, which means that he managed to prove an obvious fallacy. It seems that using his method, he can prove anything. And indeed he can. See what he is claiming when stating that the murderer is a liar: ‘If I am the murderer, then I am a liar’, which means ‘if I am the murderer then this is a lie’. In other words — ‘If I am the murderer then L is true’. And … this proves that ‘I am not the murderer.'”

An End in Sight

Write down a 0:


Now mentally “flip” this string in binary, exchanging 0s and 1s, and append this new string to the existing one:

0 1

Keep this up and you’ll get a growing string of 0s and 1s:

0 1 1 0
0 1 1 0 1 0 0 1
0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0

This is the Thue–Morse sequence, named after two of its discoverers, Axel Thue and Marston Morse. One interesting property of the sequence is that, no matter how far it’s extended, it contains no “cubes,” no instances in which some nonempty string occurs three times in a row. For example, the last line above contains both 11 and 00 but no instance of 111 or 000. (It also contains 1001 twice in a row, but not three times.)

Max Euwe, the Dutch mathematician who was world chess champion from 1935 to 1937, used this principle to show that chess was not a finite game. Under the rules at the time, a chess game would end in a draw if a sequence of moves (with all pieces in the same positions) were played three times in a row. Euwe used the Thue-Morse sequence to show that this need never happen: If 0 represents one set of moves, and 1 represents another, and each set leaves the board position unchanged, then the Thue-Morse sequence shows that two players might step through these routines forever without ever playing one three times in a row.

Modern chess rules have dropped the threefold sequence provision. Instead a draw results when the same board position occurs three times, or when 50 successive moves occur without a capture or a pawn move. Both of these rules limit a game to a finite length (although one player must actually claim the draw).

(Thanks, Pål.)


Gustave Flaubert posed this teasing problem to his sister Caroline in an 1841 letter:

Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?

He didn’t give an answer. Elsewhere he wrote, “To be stupid, selfish, and have good health are three requirements for happiness — though if stupidity is lacking, all is lost.”

Paint by Number

When Mariner 4 flew past Mars in summer 1965, NASA scientists were eager to get their first close look at another planet. So rather than wait for their computers to render the probe’s data into a proper photograph, the employees in the agency’s telecommunications group mounted printed strips of data in a display panel and colored them by hand to create a rough visualization.

The hand-colored vista became the first image of Mars based on data collected by an interplanetary probe. They framed the finished image and presented it to agency director William H. Pickering.

Double Alphamagic Squares

In 1986 British electronics engineer Lee Sallows invented the alphamagic square:

alphamagic square 1

As in an ordinary magic square, each row, column, and long diagonal produces the same sum. But when the number in each cell is replaced by the length of its English name (25 -> TWENTY-FIVE -> 10), a second magic square is produced:

alphamagic square 2

Now British computer scientist Chris Patuzzo, who found the percentage-reckoned pangram that we covered here in November 2015, has created a double alphamagic square:

double alphamagic square 1

Each row, column, and long diagonal here totals 303370120164. If the number in each cell is replaced by the letter count of its English name (using “and” after “hundred,” e.g. ONE HUNDRED AND FORTY-EIGHT BILLION SEVEN HUNDRED AND TWENTY-EIGHT MILLION THREE HUNDRED AND SEVENTY-EIGHT THOUSAND THREE HUNDRED AND SEVENTY-EIGHT), then we get a new magic square, with a common sum of 345:

double alphamagic square 2

And this is itself an alphamagic square! Replace each number with the length of its name and you get a third magic square, this one with a sum of 60:

double alphamagic square 3

Chris has found 50 distinct doubly alphamagic squares, listed here. I suppose there must be some limit to this — is a triple alphamagic square even possible?

(Thanks, Chris and Lee.)

Nature Reading
Image: Wikimedia Commons

In Germany, where modern forestry began, a curious new sort of literature arose in the 18th century:

Some enthusiast thought to go one better than the botanical volumes that merely illustrated the taxonomy of trees. Instead the books themselves were to be fabricated from their subject matter, so that the volume on Fagus, for example, the common European beech, would be bound in the bark of that tree. Its interior would contain samples of beech nuts and seeds; and its pages would literally be its leaves, the folios its feuilles.

That’s from Simon Schama’s Landscape and Memory, 1995. These xylotheques, or wood repositories, grew up throughout the developed world — the largest, now held by the U.S. Forest Service, houses 60,000 samples. “But the wooden books were not pure caprice, a nice pun on the meaning of cultivation,” Schama writes. “By paying homage to the vegetable matter from which it, and all literature, was constituted, the wooden library made a dazzling statement about the necessary union of culture and nature.”