Suppose that when Marcellus besieged Syracuse, Archimedes was standing at a corner of the city wall. A ditch runs parallel to the wall, separated from it by a distance a. To Archimedes’ left at distance b along the wall stands a catapult, which is distance c from a line perpendicular to the ditch. If Archimedes’ line of sight to the camp runs perpendicular to the wall and the ditch, show that he stood a distance ab/c from the camp.
Hungarian sculptor and architect Ernő Rubik presented this puzzle in 1988; it was originally created by Christopher C. Wiggs and Christopher J. Taylor. The puzzle has two sides, with nine clocks on each side, and the goal is to set all the clocks to 12 o’clock simultaneously.
There are two ways to adjust the clocks. Turning a wheel at any of the four corners will adjust the clock at that corner on both sides of the puzzle. And turning a wheel will also adjust the three clocks adjacent to that corner on one side of the puzzle or the other; which side is determined by the four buttons surrounding the central clock.
So, for example, pressing the northwest button “in” and then turning the northwest wheel will adjust the northwestern quartet of clocks and the corresponding corner clock on the other side of the puzzle. Pulling the northwest button “out” and turning the same wheel will adjust the northwestern clock on the front of the puzzle, its counterpart on the back, and the three clocks adjacent to it on that side.
This is more intuitive than it sounds. Here’s a simulator.
Since there are 14 independent clocks, with 12 settings each, there are a total of 1214 = 1,283,918,464,548,864 possible configurations. It turns out that no configuration requires more than 12 moves to solve; for comparison, in the “worst case” solving a Rubik’s cube can take 20 moves. The trouble, of course, is knowing how to go about it.
An old puzzle asks: Without lifting your pencil from the paper, can you draw a series of four straight lines that passes through all nine points in this grid?
The trick is to realize that the lines can extend beyond the grid’s area:
In 1970 Solomon Golomb and John Selfridge found a way to draw a closed path of eight segments that passes through all 25 points in this grid:
Can you?
A problem from the Tenth International Mathematical Olympiad, 1968:
Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.
On the morning of Jan. 20, 1953, the body of 18-year-old Paul Emanuel Rubin was found at the bottom of a ditch near the Philadelphia International Airport. The coroner found there was enough cyanide in his body to “kill 10 men,” and taped to his abdomen was a 7″ x 3″ piece of paper with an enciphered message:
Rubin’s mother hadn’t seen him since the previous morning, when he’d cut some strips of adhesive tape before leaving the house. He was studying chemistry at New York University and would have had access to cyanide, but his mother said he was in good mental and physical health and hadn’t appeared worried about anything. (About 20 minutes before the body was found, the Rev. Robert M. Anderson had wished Rubin good morning; he found him “wild-eyed” and said “he was staring straight ahead and … the pupils of his eyes were dilated.”)
A friend mentioned that Rubin had been working with codes: “They’re very complicated. Anyone who reads science fiction will know what I mean.” Rubin was carrying a copy of Galaxy Science Fiction, as well as a plastic cylinder containing a signal fuse, the casing of a spent .38 caliber bullet, a “fountain pen gun” of uncertain purpose, four keys, and 47 cents. He’d had $15 when he’d left home the previous morning.
An inquest turned up nothing, and the case was closed in March. The cipher has never been solved. The Cipher Foundation has more details about the case, as well as a link to Rubin’s FBI file (8 MB PDF). The fullest account of the case that I know is in Craig Bauer’s excellent Unsolved!: The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies (2017).
This inscription was found in a cemetery of French Franciscan friars at Dole. What does it mean?
A stranger asks you to shuffle an ordinary deck of cards and then cut it into three heaps. He’ll bet you $20 that at least one of the topmost cards is a king, queen, or jack. Should you take the bet?
