A Mathless Math Puzzle

hess bug puzzle

Richard Hess posed this problem in the Spring 1980 issue of Pi Mu Epsilon Journal. At noon on Monday, a bug departs the upper left corner, X, of a p × q rectangle and crawls within the rectangle to the diagonally opposite corner, Y, arriving there at 6 p.m. He sleeps there until noon on Tuesday, when he sets out again for X, crawling along another path within the rectangle and reaching X at 6 p.m. Prove that at some time on Tuesday the bug was no farther than p from his location at the same time on Monday.

Click for solution …

The Greatest Show

Before making his name with mobile sculpture, Alexander Calder was captivated by the circus. On a visit to Ringling Brothers and Barnum & Bailey Circus in New York City at age 27, Calder traveled about the big top with a sketchpad, drawing tightrope walkers, horseback riders, and acrobats. Using a free pass, he returned to the circus every day for two weeks, and then set out to make a toy circus of his own.

He assembled it from wire, cloth, leather, corks, pipe cleaners, string, and wood. He worked on it for six years, until he had 55 performers, and then put on circus parties for friends, playing music and introducing a ringmaster who would direct each of the acts. When it became too fragile to handle, he gave the circus to the Whitney Museum of American Art in New York City, where it remains today.

“Sandy is evidently always happy, or perhaps up to some joke, for his face is always wrapped up in that same mischievous, juvenile grin,” his school yearbook description had read. “This is certainly the index to the man’s character in this case, for he is one of the best natured fellows there is.”

Pentalpha

https://commons.wikimedia.org/wiki/File:Pentagram_green.svg

During a visit to Crete in 1938, Miss L.S. Sutherland described a game she saw played on a pentagram:

You have nine pebbles, and the aim is to get each on one of the ten spots. You put your pebble on any unoccupied spot, saying ‘one’, and then move it through another, ‘two’, whether this spot is occupied or not, to a third, ‘three’, which must be unoccupied when you reach it; these three spots must be in a straight line. If you know the trick, you can do this one-two-three trick, for each of your nine pebbles and find it a berth, and then you win your money. If you don’t know the trick, it’s extremely hard to do it.

To make this a bit clearer: The figure has 10 “spots,” the five points of the star and the five corners of the pentagon in the middle. A move consists of putting a pebble on any unoccupied spot, moving it through an adjacent spot (which may be occupied) and continuing in a straight line to the next adjacent spot, which must be unoccupied. You then leave the pebble there and start again with a new pebble, choosing any unoccupied spot to begin this next move. If you can fill 9 of the 10 spots in this way then you’ve won.

Can you find a solution?

Click for solution …

Pandigital Pi

In the July/August 2006 issue of MIT Technology Review, Richard Hess noted that this expression:

 3 + \frac{16 - 8^{-5}}{97 + 2^{4}} \approx \pi - 3.3 \times 10^{-9}

provides a good approximation to π using each of the digits 1-9 once. He challenged readers to do better, limiting themselves to the operators +, -, ×, ÷, exponents, decimal points, and parentheses.

The best solution received was from Joel Karnofsky:

 3.14 + \left ( 7^{-.9^{-6}} + 2/8 \right )^{5} \approx \pi - 9.3 \times 10^{-11}

But Karnofsky noted that this is probably not the best possible. “Unfortunately, my estimate is that there are on the order of 1016 unique values that can be generated under the given conditions and I cannot see how to avoid checking essentially all of them to fnd a guaranteed best. With maybe a thousand computers I think this could be done in my lifetime.”

Indeed, eight months later Sergey Ioffe sent this solution, which he found “using a genetic-like algorithm applying mutations to a population of parse trees and keeping some number of best ones”:

 3 + 5^{-\left ( 7^{.1} \right )} + \frac{.49^{8}}{2^{6}}  \approx \pi - 3.8 \times 10^{-13}

Even that has now been surpassed — on the Contest Center’s ongoing pi approximation page, Oleg Vlasii offers this expression:

 \left ( \frac{2}{.98} - .3 \right ) \times \left (.4 + 5^{(7^{-.6}-.1)}  \right )  \approx \pi - 4.1 \times 10^{-14}

And it’s possible to do even better than this if zero is added as a tenth digit.

Letter From Home

https://commons.wikimedia.org/wiki/File:German_POWs_lined_up_in_camp.png
Image: Wikimedia Commons

In Love Letters of the Great War (2014), Mandy Kirkby quotes this letter sent from Gleiwitz, Upper Silesia, in April 1917:

Dear Husband!

This is the last letter I am writing to you, because on the 24th I am going to marry another man. Then I don’t have to work any longer. I have already been working for three years as long as you are away from home. All the other men come home for leave, only you POWs never come. Nobody knows how long it will take until you come home. That’s why I am going to have a new husband. I will give the children to the orphanage. I don’t give a rat’s ass about a life like that! There is no way to survive with these few Pfennig benefits. At work they have a big mouth about the women. Now I don’t need to go to work, now the other man is going to work for me. All wives whose husbands are POWs will do the same thing and they will all get rid of the children. Three years at work are too much for the women and 20 Mark for benefit and 10 Mark child benefit are not enough. One cannot live on that. Everything is so expensive now. One pound of bacon costs 8 Mark, a shirt, 9 Mark.

Your wife

“We don’t know anything more about this unfortunate couple, but the strain of separation has brought the wife to breaking point,” Kirkby writes. “Whether she carried out her threat, we’ll never know.”

A Guilty Key

https://commons.wikimedia.org/wiki/File:SJTilden_of_NY.jpg

In the contentious presidential election of 1876, the campaign of Democrat Samuel Tilden sent many enciphered messages to its agents in contested states. Two years after the election, the New York Daily Tribune published some of the deciphered telegrams, showing that Tilden’s campaign had tried to bribe election officials to win the race. Here’s one of the telegrams:

https://archive.org/details/41748389078762

Since only 10 letters are used, it seems likely that the cipher refers to pairs of letters. So if each successive pair in the message is assigned to an arbitrary letter:

https://archive.org/details/41748389078762

… then we have a simple cryptogram that can be solved to give the message:

https://archive.org/details/41748389078762

Tilden’s campaign did the same thing with pairs of numbers. For example, this message:

https://archive.org/details/41748389078762

… turns out to mean:

https://archive.org/details/41748389078762

In 1879 the Tribune’s experts worked out the letter and number pairs that had corresponded to each letter of the alphabet:

https://archive.org/details/41748389078762

But it wasn’t until 1952 that cryptographer William F. Friedman reconstructed the table that the agents had used to remember this system:

https://archive.org/details/41748389078762

“It is amusing to note that the conspirators selected as their key a phrase quite in keeping with their attempted illegalities — HIS PAYMENT — for bribery seems to have played a considerable part in that campaign.”

(From Beaird Glover, Secret Ciphers of the 1876 Presidential Election, 1991.)

Podcast Episode 145: The Pied Piper of Saipan

https://commons.wikimedia.org/wiki/File:Gabaldon_1944.jpg

Guy Gabaldon was an untested Marine when he landed on the Pacific island of Saipan during World War II. But he decided to fight the war on his own terms, venturing alone into enemy territory and trying to convince Japanese soldiers to surrender voluntarily. In this week’s episode of the Futility Closet podcast we’ll follow Gabaldon’s dangerous crusade and learn its surprising results.

We’ll also examine Wonder Woman’s erotic origins and puzzle over an elusive murderer.

Intro:

In 1955 Dodge introduced the La Femme — “the first car ever exclusively designed for the woman motorist.”

In 1911 a 16-year-old English girl died when a gust of wind carried her 20 feet into the air.

Sources for our feature on Guy Gabaldon:

Guy Gabaldon, Saipan: Suicide Island, 1990.

“Diminutive WWII Hero Gabaldon Dies at 80,” Associated Press, Sept. 4, 2006.

Richard Goldstein, “Guy Gabaldon, 80, Hero of Battle of Saipan, Dies,” New York Times, Sept. 4, 2006.

Jocelyn Y. Stewart, “Guy Gabaldon, 80; WWII Hero Captured 1,000 Japanese on Saipan,” Los Angeles Times, Sept. 6, 2006.

“Guy Gabaldon,” Latino Americans, PBS, Sept. 24, 2013.

Richard Gonzalez, “Filmmaker: Pacific War Hero Deserved Higher Honor,” Morning Edition, National Public Radio, April 25, 2008.

“Guy Gabaldon: An Interview and Discussion,” War Times Journal (accessed Feb. 26, 2017).

“Milestones,” Time 168:12, Sept. 18, 2006.

Gregg K. Kakesako, “‘Pied Piper’ Returning to Saipan,” Honolulu Star Bulletin, June 6, 2004.

“Guy Gabaldon,” University of Texas Oral History Project (accessed Feb. 26, 2017).

Gabaldon receives the Navy Cross, 1960:

Listener mail:

Wikipedia, “William Moulton Marston” (accessed March 9, 2017).

“The Man Behind Wonder Woman Was Inspired By Both Suffragists And Centerfolds,” NPR Books, October 27, 2014.

Jill Lepore, “The Surprising Origin Story of Wonder Woman,” Smithsonian Magazine, October 2014.

Katha Pollitt, “Wonder Woman’s Kinky Feminist Roots,” Atlantic, November 2014.

This week’s lateral thinking puzzle was contributed by listener Steven Jones (thanks also to Hanno Zulla). Here are three corroborating links (warning — these spoil the puzzle).

You can listen using the player above, download this episode directly, or subscribe on iTunes or Google Play Music or via the RSS feed at http://feedpress.me/futilitycloset.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation on the Support Us page of the Futility Closet website.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

Gratitude

guerrero clock case

During the Great Depression, Texas carpenter Ernesto Guerrero could not find work, and his family received boxes of food through Franklin Roosevelt’s New Deal assistance program. To show her appreciation, Ernesto’s daughter Ernestine spent a year developing her skills with a coping saw and another year carving this fretwork clock case from the leftover grocery boxes.

She sent it to Roosevelt in 1937. “This is the best I have ever done in my life,” she wrote. “I know that you have many pretty things, but please accept and keep this piece of work from a poor girl that doesn’t have anything, also to show you how much we admire you … as a man of great ideals and a big heart toward humanity.”

FDR put the case on display, first at his family estate in Hyde Park, N.Y., and later as part of the Franklin D. Roosevelt Library Museum Collection — the first presidential library.

(From Robert Cohen, Dear Mrs. Roosevelt: Letters from Children of the Great Depression, 2003.)