The Ranchers’ Split

A problem from the Graham Dial, a publication of Graham Transmissions Inc., via L.A. Graham’s The Surprise Attack in Mathematical Problems (1968):

Little Euclid went with his father to visit a friend who ran a small ranch. The rancher told them that he and his partner used to herd cattle but had then sold the herd for as many dollars per head as there were head in the herd. With the proceeds they had bought as many sheep as they could at $10.00 per sheep, and with the money that remained they bought a dog. Later the rancher and his partner decided to split up evenly, so they added the dog to the herd and each of them took the same number of animals.

Euclid surprised him by saying, “I hope you paid him $2.00 to even things up.”

“Why, yes, I did,” the rancher said, “but how did you know? I haven’t told you any of the numbers or prices.” Can you prove that Little Euclid had to be correct?

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First, three observations: (1) The number of dollars that the men received for the herd of cattle was a perfect square. (2) The number of sheep they bought was odd, because it was necessary to throw in the dog in order to divide the animals evenly. (3) Since they bought an odd number of sheep at $10.00 per head, the next-to-last digit in the square payment from observation (1) must be odd. Now, as it happens, in any perfect square, if the next-to-last digit is odd, the last figure must be a 6. The simplest way to see this is to consider that any number can be expressed as 10a + b, where a comprises all the digits except the last and b is the last digit (0 to 9). The square of this number is 100a² + 20ab + b². But 100a² has zeros in the last two places, and 20ab has a zero in the last place and is preceded by an even number. So, in order for the square to have an odd digit in the next-to-last position, b² must have an odd first digit, and the only possibilities are 4² = 16 and 6² = 36. In either case, the dog cost $6.00, and the rancher owed his partner $2.00 for accepting a $6.00 dog in place of a $10.00 sheep.

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August 7, 2018 | Puzzles

Discretion

https://commons.wikimedia.org/wiki/File:Looking_across_the_Cherwell_at_Parsons_Pleasure_(geograph_5379091).jpg — Image: Wikimedia Commons

A group of Oxford dons were sunbathing nude by the River Cherwell when a female student floated by in a punt.

Most of them scrambled to cover their nakedness, but Sir Maurice Bowra instead put a flannel over his head.

When asked why, he said, “I don’t know about you, gentlemen, but in Oxford I, at least, am known by my face.”

August 7, 2018August 6, 2018 | Society

Next Up

What’s the next number in this sequence?

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, …

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The next number is 9. This is Golomb’s sequence, a non-decreasing sequence of integers in which a_n is the number of times that n occurs in the sequence. We’re given a₁ = 1 to start with, and this indicates that the number 1 appears only once in the whole sequence. That means that a₂ is greater than 1. Setting a₂ to 2 immediately tells us that 2 appears twice in the sequence, so a₃ also must be 2. That tells us that 3 occurs twice in the sequence, and so on. In this way each number in the sequence is uniquely determined.

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August 6, 2018 | Puzzles

Podcast Episode 211: Cast Away on an Ice Floe

Germany’s polar expedition of 1869 took a dramatic turn when 14 men were shipwrecked on an ice floe off the eastern coast of Greenland. As the frozen island carried them slowly toward settlements in the south, it began to break apart beneath them. In this week’s episode of the Futility Closet podcast we’ll follow the crew of the Hansa on their desperate journey toward civilization.

We’ll also honor a slime mold and puzzle over a reversing sunset.

See full show notes …

August 6, 2018August 13, 2018 | History · Podcast

Jack the Snipper

From the United Press, June 18, 1942:

Pascagoula, Miss. — (U.P.) — Everybody in town is just as mystified over the motive of the ‘phantom barber’ as they are about who he might try to clip next.

Without robbing or otherwise disturbing his victims, he breaks into homes at night and snips the hair of heavy sleepers. He has given haircuts to three persons in the past week and not one of them even woke up during the process.

Police chief A. W. Ezell said he didn’t have the slightest idea why a man would want to do such a thing, but because the complaints have been coming hard and heavy, his department has staked a $300 reward for information leading to his capture. He also gave pistol permits to six volunteer officers and ordered the regular police force to be on the alert.

Bloodhounds, given a man’s footprint to start on, have failed miserably. None of the victims could give a description since they slumbered on oblivious of the tonsorial attention they were getting.

August 5, 2018August 4, 2018 | Crime · Oddities

Custom Baking

From a Russian puzzle collection:

Is it possible to bake a cake that can be divided into four parts by a single straight cut?

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Yes — here’s one possibility: (From E.G. Kozlova, Skazki i podskazki: Zadachi dlya matematicheskogo kruzhka [Fairy Tales and Hints: Problems for a Mathematical Circle], 2004 [in Russian], via Jennifer Beineke and Jason Rosenhouse, The Mathematics of Various Entertaining Subjects, 2017.)

August 4, 2018 | Puzzles

Stuckie

Loggers with the Georgia Kraft Corp. were cutting down a chestnut oak in southern Georgia in the 1980s when they discovered the mummified remains of a dog inside the hollow trunk. Experts determined that it was most likely a hunting dog that had pursued some quarry up through the hollow tree sometime in the 1960s. The dog had wedged itself into the narrowing trunk and, unable to turn around, eventually perished 28 feet above ground level. But chestnut oak contains tannin, a natural desiccant that stopped microbial activity, and the dog’s position and a natural updraft through the trunk prevented other animals from scenting or reaching it. So it just remained there for 20 years, waiting to be found.

The mummified dog, now known as “Stuckie,” is on display at the Southern Forest World museum in Waycross, Georgia.

August 3, 2018August 3, 2018 | Death · Oddities

Unquote

“I am reminded of the professor who, in his declining hours, was asked by his devoted pupils for his final counsel. He replied, ‘Verify your quotations.'” — Winston Churchill

August 3, 2018August 2, 2018 | Quotations

Swahili Time

https://commons.wikimedia.org/wiki/File:Red_sky_%2B_Giraffe_%2B_Kenya_%3D_A_sunrise_to_remember_(15062623745).jpg — Image: Wikimedia Commons

Kenya and Uganda both lie on the equator, so the sun rises around 6 a.m. and sets around 6 p.m. throughout the year. Given such a reliable natural timekeeper, it’s customary to reckon time by counting hours of light or hours of darkness: 7 a.m. is called 1 o’clock (saa moja, or one hour of light), and 11 a.m. is called 5 o’clock (saa tano) (moja means 1 and tano 5 in Swahili). Similarly, 7 p.m. is called 1 o’clock (one hour of darkness), and 11 p.m. is 5 o’clock.

Confusingly for newcomers, clocks themselves are set to Western time, but they’re read aloud in “Swahili time.” Increasingly, though, Africans are simply conforming to Western conventions.

August 2, 2018 | Science & Math