Can every convex polyhedron be “unfolded” into a shape that doesn’t overlap itself, like this dodecahedron?
Surprisingly, no one knows.
The Renaissance artist Albrecht Dürer first wondered about this in the early 1500s, and in the ensuing five centuries no one has found a polyhedron that won’t permit it. But one may yet exist.
A problem from Sam Loyd’s Cyclopedia of Puzzles, 1914:
Here is the puzzle of Tom the Piper’s Son, who, as told by ‘Mother Goose,’ stole the pig and away he run. It is known that Tom entered the far gate shown at the top on the right hand. The pig was rooting at the base of the tree 250 yards distant, and Tom captured it by always running directly towards it, while the pig made a bee-line towards the lower corner as shown. Now, assuming that Tom ran one-third faster than the pig, how far did the pig run before he was caught?
Intriguingly, Loyd adds, “The puzzle is a remarkable one on account of its apparent simplicity and yet the ordinary manner of handling problems of this character is so complicated that solvers are asked merely to submit approximately correct answers, based upon judgment and common sense, just to see who can make the best guess. The simple rule for solving it, however, which will doubtless be quite new to our puzzlists, is based upon elementary arithmetic.” What’s the answer?
Square roots:
EIGHTY-ONE has 9 letters.
ONE HUNDRED has 10 letters.
FIVE HUNDRED AND SEVENTY SIX has 24 letters.
Cube roots:
THIRTY-NINE THOUSAND THREE HUNDRED FOUR has 34 letters.
SIXTY-EIGHT THOUSAND NINE HUNDRED AND TWENTY-ONE has 41 letters.
ONE MILLION and ONE BILLION have 10 letters each, making them a sixth root and (in the United States) a ninth root word.
(Dave Morice, “Kickshaws,” Word Ways 30:2 [May 1997], 129-141.)
10/26/2020 UPDATE: Reader Hans Havermann has found many more, including this alarming specimen:
341183 = ONE HUNDRED SIXTY-NINE SENONAGINTILLION, FIVE HUNDRED FIVE QUINONAGINTILLION, SEVENTY-SEVEN QUATTUORNONAGINTILLION, FIFTY-ONE TRENONAGINTILLION, EIGHT HUNDRED SEVENTY DUONONAGINTILLION, SEVEN HUNDRED FIFTY-FOUR UNONAGINTILLION, SEVEN HUNDRED THIRTY-EIGHT NONAGINTILLION, SIX HUNDRED FIFTY-SIX NOVOCTOGINTILLION, TWO HUNDRED SEVENTY OCTOCTOGINTILLION, TWO HUNDRED NINETY-EIGHT SEPTENOCTOGINTILLION, SIX HUNDRED FORTY-SEVEN SEXOCTOGINTILLION, EIGHT HUNDRED FORTY-EIGHT QUINTOCTOGINTILLION, FOUR HUNDRED FOUR QUATTUOROCTOGINTILLION, EIGHT HUNDRED FIFTY-SIX TRESOCTOGINTILLION, SIX HUNDRED THIRTEEN DUOOCTOGINTILLION, EIGHT HUNDRED THIRTY-THREE UNOCTOGINTILLION, NINE OCTOGINTILLION, SEVEN HUNDRED SIXTY-FIVE NOVEMSEPTUAGINTILLION, EIGHT HUNDRED SIXTY-SEVEN OCTOSEPTUAGINTILLION, ONE HUNDRED FIFTY-THREE SEPTENSEPTUAGINTILLION, NINE HUNDRED SEVENTY-TWO SESEPTUAGINTILLION, EIGHT HUNDRED SEVENTY-EIGHT QUINSEPTUAGINTILLION, FIFTY-NINE QUATTUORSEPTUAGINTILLION, SEVEN HUNDRED SIX TRESEPTUAGINTILLION, EIGHT HUNDRED FIFTEEN DUOSEPTUAGINTILLION, TWO HUNDRED THIRTY-ONE UNSEPTUAGINTILLION, FOUR HUNDRED EIGHT SEPTUAGINTILLION, ONE HUNDRED FIFTY-FIVE NOVEMSEXAGINTILLION, FOUR HUNDRED EIGHTY-SIX OCTOSEXAGINTILLION, TWELVE SEPTENSEXAGINTILLION, TWO HUNDRED FORTY-SIX SESEXAGINTILLION, EIGHT HUNDRED EIGHTY-FIVE QUINSEXAGINTILLION, TWO HUNDRED FIFTY-SIX QUATTUORSEXAGINTILLION, FOUR HUNDRED NINETY-TWO TRESEXAGINTILLION, SEVEN HUNDRED FORTY-THREE DUOSEXAGINTILLION, FIVE HUNDRED FIFTY-FIVE UNSEXAGINTILLION, ONE HUNDRED FORTY-ONE SEXAGINTILLION, EIGHT HUNDRED TWENTY-FIVE NOVEMQUINQUAGINTILLION, SEVEN HUNDRED FORTY-SIX OCTOQUINQUAGINTILLION, ONE HUNDRED SEVENTY-ONE SEPTENQUINQUAGINTILLION, NINE HUNDRED THIRTY-TWO SEXQUINQUAGINTILLION, ONE HUNDRED SIXTY-SIX QUINQUINQUAGINTILLION, FOUR HUNDRED THREE QUATTUORQUINQUAGINTILLION, THIRTY-TWO TRESQUINQUAGINTILLION, SIX HUNDRED TWENTY-FOUR DUOQUINQUAGINTILLION, ONE HUNDRED FORTY-SIX UNQUINQUAGINTILLION, ONE HUNDRED TWENTY-FIVE QUINQUAGINTILLION, EIGHT HUNDRED TWO NOVEMQUADRAGINTILLION, EIGHT HUNDRED FIFTY-EIGHT OCTOQUADRAGINTILLION, ONE HUNDRED SEVENTY-NINE SEPTENQUADRAGINTILLION, ONE HUNDRED FIFTY-SIX SEXQUADRAGINTILLION, ONE HUNDRED TWENTY-FOUR QUINQUADRAGINTILLION, NINE HUNDRED FIFTY-FOUR QUATTUORQUADRAGINTILLION, NINE HUNDRED ELEVEN TRESQUADRAGINTILLION, TWO HUNDRED FIFTEEN DUOQUADRAGINTILLION, NINE HUNDRED SIXTY-SIX UNQUADRAGINTILLION, NINE HUNDRED TWELVE QUADRAGINTILLION, EIGHT HUNDRED THREE NOVEMTRIGINTILLION, SIX HUNDRED SIXTY-EIGHT OCTOTRIGINTILLION, SEVEN HUNDRED SIXTY-THREE SEPTRIGINTILLION, ONE HUNDRED EIGHTY-THREE SEXTRIGINTILLION, SEVEN HUNDRED NINETY-SIX QUINTRIGINTILLION, NINETY QUATTUORTRIGINTILLION, SEVEN HUNDRED TRESTRIGINTILLION, FIVE HUNDRED EIGHTY-SIX DUOTRIGINTILLION, ONE HUNDRED FORTY-ONE UNTRIGINTILLION, SIX HUNDRED FIFTY-FOUR TRIGINTILLION, SIX HUNDRED EIGHT NOVEMVIGINTILLION, SEVEN HUNDRED FIFTY-THREE OCTOVIGINTILLION, SIX HUNDRED FOURTEEN SEPTENVIGINTILLION, THREE HUNDRED FOUR SEXVIGINTILLION, SEVEN HUNDRED NINETY-FIVE QUINVIGINTILLION, SIX HUNDRED FORTY-FOUR QUATTUORVIGINTILLION, THREE HUNDRED SIXTY-FIVE TREVIGINTILLION, EIGHT HUNDRED NINETY-SEVEN DUOVIGINTILLION, EIGHT HUNDRED SEVENTEEN UNVIGINTILLION, FIVE HUNDRED EIGHTY-TWO VIGINTILLION, FOUR HUNDRED FIFTY-FIVE NOVEMDECILLION, FOUR OCTODECILLION, SIX HUNDRED FORTY-FIVE SEPTENDECILLION, NINE HUNDRED SEVEN SEXDECILLION, FOUR HUNDRED EIGHTY QUINDECILLION, EIGHT HUNDRED THIRTY-SEVEN QUATTUORDECILLION, ONE HUNDRED FORTY-FOUR TREDECILLION, TWO HUNDRED TWENTY-FIVE DUODECILLION, EIGHT HUNDRED EIGHTY-ONE UNDECILLION, SIX HUNDRED SEVEN DECILLION, THREE HUNDRED FIFTY-TWO NONILLION, FIVE HUNDRED FORTY-THREE OCTILLION, FOUR HUNDRED NINE SEPTILLION, TWO HUNDRED SEVENTY-NINE SEXTILLION, ONE HUNDRED EIGHTY-EIGHT QUINTILLION, SEVEN HUNDRED SEVENTY-SIX QUADRILLION, FORTY-EIGHT TRILLION, THREE HUNDRED NINETY-FOUR BILLION, SEVEN HUNDRED SEVENTY-ONE MILLION, EIGHT HUNDRED EIGHT THOUSAND, THREE HUNDRED THIRTY-ONE
The name of that number contains 3,411 letters.
(Thanks, Hans.)
The Society of Actuaries holds a regular speculative fiction contest. Here’s an excerpt from “The Temple of Screens,” by Nate Worrell, FSA, MAAA, recognized last year for describing the “most innovative actuarial career of the future”:
‘Ever since humans began to be aware of a future, we’ve wanted to explore it. We’ve cast stones, searched in tea leaves, held the entrails of animals in our hands to try to extract some knowledge of our fate. Some of our stories try to show us that like Oedipus, we can’t change our fate. In other stories, we find an escape, we have the power of choice, at least to some degree. But in either case, knowing our future changes how we act. Now that you’ve seen your possible futures, they are tainted. If you were to go back in, they’d all change, reflecting that you had some knowledge. The algorithm would reallocate a new set of weights to your tendencies, increasing some behaviors and decreasing others.’
A wave of anger flashes through me, and I stand and start pacing. ‘So what’s the use of this?’
‘To help you embrace what’s possible, to come to terms with it. You came here because you were afraid of a certain future, one you hoped to avoid somehow. We can’t fight or flee from the future, whatever one we fall into. But we can find serenity in any of our futures, if we so desire.’
The angle of Earth’s axis varies between 22.1 and 24.5 degrees over a 41,000-year period. This means that the Tropics of Cancer and Capricorn are moving slightly: Each is the most extreme circle of latitude in its hemisphere at which the Sun can be directly overhead. At the moment Cancer is drifting southward and Capricorn northward, each at about 15 meters a year.
In Mexico this movement is reflected precisely in a series of annual markers beside Federal Highway 83, from 2005 to 2010.
(Thanks, Salvador.)
Chess includes a couple of rules intended to keep a game from running on forever. Specifically, a game is a draw (a) if the same position occurs five times or (b) if each player makes a series of 75 moves without a capture or a pawn move. (The more familiar “threefold repetition” and “50-move” rules describe circumstances in which a player can claim a draw but isn’t obliged to.)
At this year’s SIGBOVIK, the tongue-in-cheek scientific conference named after fictional student Harry C. Bovik, Carnegie Mellon’s Tom Murphy VII presented a legal game that carefully skirts these rules to run on as long as possible — 17,697 half-moves, enough to fill 6 pages of the conference proceedings even in small type.
“It can also be downloaded at tom7.org/chess/longest.pgn. Many chess programs fail to load the whole game, but this is because they decided not to implement the full glory of chess.”
(Tom Murphy VII, “Is This the Longest Chess Game?”, SIGBOVIK 2020, Carnegie Mellon University, April 1, 2020.) (Thanks, Noëlle.)
09/29/2020 UPDATE: Reader Alexander Bolton has set up a Longest Chess Game Bot on Twitter that’s playing through this game, tweeting out an image of the position after every halfmove. “It tweets every 4 hours so it should be finished in just over 8 years!”
Biologist Jonathan Eisen, who coined the term phylogenomics, called this “perhaps the best genomics Venn diagram ever.” The six-set diagram, published by Angélique D’Hont and her colleagues in Nature in 2012, presents the number of gene families that the banana shares with five other species.
“What the diagram says is that over time the 7,674 gene clusters shared by the six species did not change much in these lineages, as opposed to the 759 clusters specific to the banana (Musa acuminata), for example,” explains Anne Vézina at ProMusa. “Although the genes in these clusters probably share common ancestors with other species, they have since changed to the point that they haven taken on new functions.”
Here’s a similar (5-set) diagram relating to conifers.
(Angélique D’Hont et al., “The Banana (Musa acuminata) Genome and the Evolution of Monocotyledonous Plants,” Nature 488:7410 [2012], 213-217.) (Thanks, David.)
Crockett Johnson, author of the 1955 children’s book Harold and the Purple Crayon, was trained as an engineer and produced more than 100 paintings based on diagrams used in the proofs of classical theorems. This one, Polar Line of a Point and a Circle (Apollonius), appears to have been inspired by a figure in Nathan A. Court’s 1966 College Geometry. The two circles are orthogonal: They cut one another at right angles. And as the square of the line connecting their centers equals the sum of the squares of their radii, these three segments form a right triangle.
Johnson was inspired to this work by his admiration of classical Greek architecture. Sitting in a restaurant in Syracuse in 1973, he managed to construct a heptagon using seven toothpicks and the edges of a menu and a wine list, a construction that had eluded the Greeks. (He found later that Archibald Finlay had illustrated similar constructions in 1959.)
(Stephanie Cawthorne and Judy Green, “Harold and the Purple Heptagon,” Math Horizons 17:1 [2009], 5-9.)
Dan Pearcy’s Twitter feed highlights a pretty link between Pascal’s triangle and the tangent angle formulae:
In the 1928 film Steamboat Bill, Jr., a falling facade threatens to flatten Buster Keaton, but he’s spared by the fortunate placement of an open attic window. “As he stood in the studio street waiting for a building to crash on him, he noticed that some of the electricians and extras were praying,” writes Marion Meade in Cut to the Chase, her biography of Keaton. “Afterward, he would call the stunt one of his greatest thrills.”
It’s often said that the falling wall missed Keaton by inches. Is that true? James Metz studied the problem in Mathematics Teacher in 2019. Keaton was 5 feet 5 inches tall; if that the “hinge” of the facade is 5 inches above the surface of the ground, the attic window is 12 feet above that, and the window is 3 feet high, he finds that the top of the window came only within about 1.5 feet of Keaton’s head.
“The window was tall enough to allow an ample margin of safety, so the legend about barely missing his head cannot be true,” Metz writes. “Apparently, Keaton had more headroom than was previously suspected.”
(James Metz, “The Right Place at the Right Time,” Mathematics Teacher 112:4 [January/February 2019], 247-249.)
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