Join two of the star’s adjacent vertices, such as A and E. Triangles BMD and AME have an angle in common (the congruent angles at M), so the sum of angles B and D in triangle BMD equals the sum of angles A and E in triangle AME. This means that the sum of the star’s five angles equals the sum of the angles of triangle ACE — which is 180°.
He told the Telegraph, “Nobody significant died that day, no major events apparently occurred and, although a typical day in the 20th century has many notable people being born, for some reason that day had only one who might make that claim — Abdullah Atalar, a Turkish academic.
“The irony is, though, that — having done the calculation — the day is interesting for being exceptionally boring. Unless, that is, you are Abdullah Atalar.”
The clock face on the Marienkirche in Bergen auf Rügen, Germany, has 61 minutes. Does this mean time moves more slowly there — or more quickly?
To ensure quiet, poet Amy Lowell hired five rooms at every hotel — her own and those on either side, above, and below.
A perplexing sentence from a letter by Dorothy Osborne, describing shepherdesses in Bedfordshire, May 1653: “They want nothing to make them the happiest people in the world but the knowledge that they are so.”
OVEREFFUSIVE is a palindrome in Scrabble — its letter values are 141114411141. (Discovered by Susan Thorpe.)
The sum of the digits of every multiple of 2739726 up to the 72nd is 36. (E.M. Langley, Mathematical Gazette, 1896)
I’ll bet I have more money in my pocket than you do. (Of course I do — you have no money in my pocket!)
In 1996 a model airplane enthusiast was operating a remote-controlled plane in Phoenix Park in Dublin when the receiver died and the plane flew off on its own. It flew five miles to the northeast, ran out of fuel, and glided to a landing … on the taxi-way to Runway 28 at Dublin Airport.
A memorably phrased puzzle from The Graham Dial: “Consider a vertical girl whose waist is circular, not smooth, and temporarily at rest. Around the waist rotates a hula hoop of twice its diameter. Show that after one revolution of the hoop, the point originally in contact with the girl has traveled a distance equal to the perimeter of a square circumscribing the girl’s waist.”
Hold the hoop steady and let the girl roll around inside it:
Since the ratio of the diameters is 2:1, so is the ratio of the circumferences. This means that a point on the girl oscillates back and forth between two opposite points on the hoop, passing through the hoop’s center on the way and producing a straight line (the “Tusi couple”).
“The perimeter of circumscribing square equals four girl diameters or two hula hoop diameters which is the total displacement of initial point of contact between hula hoop and the aforementioned vertical girl.”
From L.A. Graham, The Surprise Attack in Mathematical Problems, 1968.
Frightened villagers “killed” the first hydrogen balloon, launched in Paris by Jacques Charles and the Robert brothers Anne-Jean and Nicolas-Louis on Aug. 27, 1783. Allen Andrews, in Back to the Drawing Board: The Evolution of Flying Machines, quotes a contemporary account:
It is presumed that it was carried to a height of more than 20,000 feet, when it burst by the reaction of the Inflammable Gas upon the Atmospheric Air. It fell at three quarters past five near Gonesse, ten miles [actually, 15 miles] from the Field of Mars. The affrightened inhabitants ran together, appalled by the Hellish stench of sulphur, and two monks having assured them it was the skin of a Monstrous Animal, they attacked it with stones, pitchforks and flails. The Curate of the village was obliged to attend in order to sprinkle it with holy water and remove the fears of his astonished parishioners. At last they tied to the tail of a horse the first Instrument that was ever made for an Experiment in Natural Philosophy, and trained it across the field more than 6000 feet.
In 1943 three men came up with an ingenious plan to escape from the seemingly escape-proof Stalag Luft III prison camp in Germany. In this episode of the Futility Closet podcast we’ll learn about their clever deception, which made them briefly famous around the world.
We’ll also hear about the chaotic annual tradition of Moving Day in several North American cities and puzzle over how a severely injured hiker beats his wife back to their RV.
Sources for our feature on the escape from Stalag Luft III:
It became the third most popular film at the British box office in 1950. The book’s success led Williams to write The Tunnel, a prequel that described his and Michael Codner’s earlier escape from the Oflag XXI-B camp in Poland.
This week’s lateral thinking puzzle was contributed by listener David White and his daughter Katherine.
This episode is sponsored by our patrons and by The Great Courses — go to http://www.thegreatcourses.com/closet to order from eight of their best-selling courses at up to 80 percent off the original price.
Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.
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Many thanks to Doug Ross for the music in this episode.
University of Strathclyde mathematician Adam McBride recalls that in his student days a particular teacher used to present a weekly puzzle. One of these baffled him:
Find positive integers a, b, and c, all different, such that a3 + b3 = c4.
“The previous puzzles had been relatively easy but this one had me stumped,” he wrote later. He created three columns headed a3, b3, and c4 and spent hours looking for a sum that would work. On the night before the deadline, he found one: 703 + 1053 = 354.
“This shows how sad a person I was! However, I then realised also how stupid I had been. I had totally missed the necessary insight.” What was it?
1. A puzzle from J.A.H. Hunter’s Fun With Figures, 1956:
Tom and Tim are brothers; their combined ages make up seventeen years. When Tom was as old as Tim was when Tim was twice as old as Tom was when Tom was fifteen years younger than Tim will be when Tim is twice his present age, Tom was two years younger than Tim was when Tim was three years older than Tom was when Tom was a third as old as Tim was when Tim was a year older than Tom was seven years ago. So how old is Tim?
2. Another, by Sam Loyd:
“How fast those children grow!” remarked Grandpa. “Tommy is now twice as old as Maggie was when Tommy was six years older than Maggie is now, and when Maggie is six years older than Tommy is now their combined ages will equal their mother’s age then, although she is now but forty-six.” How old is Maggie?
3. According to Wirt Howe’s New York at the Turn of the Century, 1899-1916, this question inspired an ongoing national debate when it appeared in the New York Press in 1903:
Brooklyn, October 12
Mary is 24 years old. She is twice as old as Anne was when she was as old as Anne is now. How old is Anne now? A says the answer is 16; B says 12. Which is correct?