You have 10 stacks of silver dollars, with 10 coins in each stack. The coins appear identical, but you know that all the coins in one stack are counterfeit. You know the weight of a genuine coin, and you know that a counterfeit coin weighs 1 gram less than this. How many weighings must you do to find the counterfeit stack?
Marital Duels
In the Middle Ages, husbands and wives would sometimes settle their differences with physical combat. To compensate for the man’s greater strength, his wife was given certain advantages:
The woman must be so prepared that a sleeve of her chemise extend a small ell beyond her hand like a little sack. There indeed is put a stone weighing three pounds; and she has nothing else but her chemise, and that is bound together between the legs with a lace. Then the man makes himself ready in the pit over against his wife. He is buried therein up to the girdle, and one hand is bound at the elbow to the side.
In other drawings the man sits in a tub; in one the two fight with drawn swords. “Judicial duels were common enough in the medieval and early modern period to merit etiquette books,” writes scholar Allison Coudert, “but, as far as I know, nowhere except in the Holy Roman Empire were judicial duels ever considered fitting means to settle marital disputes, and no record of such a duel has been found after 1200, at which time a couple is reported to have fought with the sanction of the civic authorities at Bâle.” The drawings that have survived come from historical treatises of the 15th and 16th centuries.
(Allison Coudert, “Judicial Duels Between Husbands and Wives,” Notes in the History of Art 4:4 [Summer 1985], 27-30.)
I Got a Feeling
My teenage children are mad about rock ‘n’ roll. I don’t mind, but between them they have socks, pullovers and slacks which are fluorescent, and I am worried in case these are harmful to their health. Surely things that are luminous in the dark are usually radioactive, which, I take it, could be dangerous.
You’ll be relieved to know that these clothes, so popular with teenagers (particularly the rock ‘n’ rollers), have been tested for radioactivity, and there is none. So there should be no danger at all, except to anyone who is sensitive to the kinds of colours they select!
— Woman’s Realm, April 12, 1958
Unquote
In order that life should be a story or romance to us, it is necessary that a great part of it, at any rate, should be settled for us without our permission. … A man has control over many things in his life; he has control over enough things to be the hero of a novel. But if he had control over everything, there would be so much hero that there would be no novel.
— G.K. Chesterton, Heretics, 1906
Podcast Episode 165: A Case of Mistaken Identity
In 1896, Adolf Beck found himself caught up in a senseless legal nightmare: Twelve women from around London insisted that he’d deceived them and stolen their cash and jewelry. In this week’s episode of the Futility Closet podcast we’ll follow Beck’s incredible ordeal, which ignited a scandal and inspired historic reforms in the English justice system.
We’ll also covet some noble socks and puzzle over a numerical sacking.
The Susquehanna Ice Bridge
In the 1850s, railroad passengers traveling from Baltimore to Philadelphia would debark at the Susquehanna River, cross the river on a ferryboat, and board a train waiting on the other side. In the severe winter of 1852, so much floating ice had piled up at this point that the ferry couldn’t be used, so railroad engineer Isaac R. Trimble came up with a novel solution: He built a railway across the ice for the baggage and freight cars, and a sledge road beside it along which horses could draw his passengers. The cars had to descend 10 to 15 feet from the bank to the surface of the ice, and at the other side they were tied to a locomotive and pulled up. The “ice bridge” opened on Jan. 15 and was accounted a great success. The Franklin Journal reported, “Forty freight cars per day, laden with valuable merchandise, have been worked over this novel tract by the means above referred to, and were propelled across the ice portion by two-horse sleds running upon the sledge road, and drawing the cars by a lateral towing line, of the size of a man’s finger.”
“At the present writing, this novel and effectual means of maintaining the communication at Havre de Grace is still in successful operation, and will so continue until the ice in the river is about to break up. Then, by means of the sledges, the rails (the only valuable part of the track), can be rapidly moved off by horse power, not probably requiring more than a few hours’ time, so that the communication may be maintained successfully until the last moment. If properly timed, as it doubtless will be, the railroad may be removed, the ice may run out, and the ferry be resumed, it may be, in less than forty-eight hours.”
In fact, wrote historian Charles P. Dare, the ice bridge operated until Feb. 24, “when it was taken up, and, in a few days, the river was free of ice. During this time, 1378 cars loaded with mails, baggage and freight were transported upon this natural bridge, the tonnage amounting to about 10,000 tons. The whole was accomplished without accident of any kind; and the materials were all removed prior to the breaking up of the river without the loss of a cross-tie or bar of iron.”
Figuring
An eight-digit number contains two 1s, two 2s, two 3s, and two 4s. The 1s are separated by 1 digit, the 2s by 2 digits, the 3s by 3 digits, and the 4s by 4 digits. What is the number?
Black and White
Antipodes
It used to be said that the sun never set on the British Empire; it still doesn’t set on the Commonwealth of Nations, as Gibraltar is on the opposite side of the earth from New Zealand’s Te Arai Beach.
At their height, the sun didn’t set on the French Empire, the Dutch Empire, or the Spanish Empire, either. New Caledonia, an overseas territory of France, is opposite Mauritania, once part of French West Africa. Parts of Suriname, a former Dutch colony, are opposite the Indonesian island Sulawesi, once part of the Netherlands East Indies. And Luzon, the largest island of the Philippines, is opposite eastern Bolivia; both were once controlled by Spain.
The Pacific Ocean is so large that it stretches more than halfway around the world — parts of the ocean are on opposite sides of the earth.
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.