Shifting Gears

A puzzle by French puzzle maven Pierre Berloquin:

Timothy rides a bicycle on a road that has four parts of equal length.

The first fourth is level, and he pedals at 10 kph.

The second fourth is uphill, and he pedals at 5 kph.

The third fourth is downhill, and he rides at 30 kph.

The fourth fourth is level again, but he has the wind at his back, so he goes 15 kph.

What is his average speed?

Click for solution …


Image: Flickr

We’ve been making things awfully hard on spirits. The standard Ouija board lays out the alphabet in two simple rows, which means it’s easy for the dead to tell us about FEEDERS but terribly hard to refer to LAYAWAY, even though these words are equally long.

In the interests of better communication, Eric Iverson made a study of this for the August 2005 issue of Word Ways. Using an image of a Ouija board, he counted the number of pixels that a planchette would have to travel in order to spell out various English words. The results are dismaying: The most exhausting four-letter word, MAMA, requires fully 17 times as much travel as the simple FEED. Longer words are more consistent: The hardest 23-letter word, DISESTABLISHMENTARIANISM, requires little more work than the easiest, ELECTROENCEPHALOGRAPHIC. But do dead people have that kind of stamina?

What’s the answer? Iverson experimented with different layouts and found a hexagonal grid that minimizes the average travel distance for a typical word (see the link below). And he found a checkerboard grid that’s 3 percent more efficient than that. Even rearranging the letters on a standard board to ZXVGINAROFUPQ JKWCHTESDLMYB rather than the standard alphabet increases efficiency by about a third. Now maybe we can have some better conversations.

(Eric Iverson, “Traveling Around the Ouija Board,” Word Ways 38:3 [August 2005], 174-177.)

The Battle of the Reed Rules

Until 1890, the minority party in the U.S. House of Representatives could block a vote by “disappearing”; they’d demand a roll call, remain silent when called upon, and then declare that too few members were “present” for the House to conduct its business.

To incoming speaker Thomas Brackett Reed this was a “tyranny of the minority,” and on Jan. 28 he resolved to break it. When Democrats demanded a roll call and refused to answer to their names, Reed marked them present anyway; when Kentucky representative James B. McCreary objected, Reed said sweetly, “The Chair is making a statement of fact that the gentleman from Kentucky is present. Does he deny it?”

There followed a sort of ontological shooting gallery. Democrats hid under their desks and behind screens to avoid being observed to exist. When they tried to flee the chamber entirely, Reed ordered the doors locked, which started a scramble to get out before the next vote. Representative Kilgore of Texas had to kick open a locked door to escape. Amid the howled objections, Confederate general “Fighting Joe” Wheeler came down from the rear “leaping from desk to desk as an ibex leaps from crag to crag,” and one unnamed Texas Democrat “sat in his seat significantly whetting a bowie knife on his boot.” Finally the Republicans mustered a majority even with the Democrats entirely absent, and the battle was over: Reed’s new rules were adopted on February 14.

Throughout all this Reed had seemed imperturbable, “serene as a summer morning.” He told a friend later that he had made up his mind what he would do if the House did not support him. “I would simply have left the Chair and resigned the Speakership and my seat in Congress,” he said. “I had made up my mind that if political life consisted in sitting helplessly in the Speaker’s Chair and seeing the majority helpless to pass legislation, I had had enough of it and was ready to step down and out.”

(From Barbara Tuchman’s The Proud Tower.) (Thanks, Zach.)



“Murder is a crime. Describing murder is not. Sex is not a crime. Describing sex is.” — Gershon Legman

Podcast Episode 76: Get Out of Jail Free

During World War II, the British Secret Service found a surprising way to help Allies in Nazi prisoner-of-war camps: They used doctored Monopoly sets to smuggle in maps, files, compasses, and real money. In this week’s episode of the Futility Closet podcast we’ll tell the story behind this clever ploy, which may have helped thousands of prisoners escape from Nazi camps.

We’ll also hear listeners’ thoughts on Jeremy Bentham’s head, Victorian tattoos, and phone-book-destroying German pirates and puzzle over murderous cabbies and moviegoers.

Sources for our feature on MI9’s use of Monopoly sets to help Allied prisoners escape during World War II:

Philip E. Orbanes, Monopoly: The World’s Most Famous Game — And How It Got That Way, 2006.

Ki Mae Heussner, “Get Out of Jail Free: Monopoly’s Hidden Maps,” ABC News, Sept. 18, 2009 (retrieved Sept. 27, 2015).

Listener mail:

Myths and legends surrounding Jeremy Bentham’s auto-icon, from University College London.

This week’s lateral thinking puzzles are from Matthew Johnstone’s 1999 book What’s the Story?

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at

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.

You can also make a one-time donation via the Donate button in the sidebar 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 Thanks for listening!

Easy Pi

Here’s a simple algorithm that Yoshiaki Tamura and Yasumasa Kanada used to calculate π to 16 million places. It’s based on Gauss’ study of the arithmetic-geometric mean of two numbers. “Instead of using an infinite sum or product, the calculation goes round and round in a loop,” writes David Wells in The Penguin Dictionary of Curious and Interesting Numbers. “It has the amazing property that the number of correct digits approximately doubles with each circuit of the loop.” Start with these values:


Then follow these instructions:

\textrm{Let}\:  \mathrm{Y}=\mathrm{A}

\textrm{Let}\:  \mathrm{A}=\displaystyle\frac{\mathrm{A}+\mathrm{B}}{2}

\textrm{Let}\:  \mathrm{B}=\sqrt{\mathrm{BY}}

\textrm{Let}\:  \mathrm{C}=\mathrm{C}-\mathrm{X}(\mathrm{A}-\mathrm{Y})^{2}

\textrm{Let}\:  \mathrm{X}=2\mathrm{X}

\textrm{PRINT}\: \displaystyle\frac{\left ( {\mathrm{A}+\mathrm{B}} \right )^{2}}{{4\mathrm{C}}}

The last instruction prints the first approximation to π; then you loop up to the top and run through the instructions again.

Running through the loop just three times gives an approximation to π that’s already correct to 5 decimal places:

Loop 1: 2.9142135
Loop 2: 3.1405797
Loop 3: 3.1415928

And running the loop a mere 19 times gives π correct to more than 1 million decimal places.


This website is supported mostly by readers’ donations. You can help by pledging a monthly contribution to our Patreon campaign. You choose the amount to contribute, and you can change or cancel your pledge at any time. In return we offer bonus posts and other rewards — see the page for details.

If you prefer, there’s a separate Patreon campaign, with different rewards, for our podcast. You can also make a one-time donation via the Donate button in the sidebar. My sincere thanks to everyone who’s contributed — we couldn’t continue without you.

I also want to thank Dan Farrow for his recent help in fixing up the website’s design and Daniel Summers for his ceaseless reliability in administering the server.

Worldly Wise

Proverbs from around the world:

  • A pretty basket does not prevent worries. (Congo)
  • Good painters need not give a name to their pictures; bad ones must. (Poland)
  • Sickness comes riding on horseback and goes away on foot. (Belgium)
  • The spectator is a great hero. (Afghanistan)
  • Those who have to go ten miles must regard nine as only halfway. (Germany)
  • The world is dark an inch ahead. (Japan)
  • Those who place their ladder too steeply will easily fall backward. (Czech Republic)
  • All the wealth of the world is in the weather. (Scotland)
  • Those whose mother is naked are not likely to clothe their aunt. (Sudan)
  • To be in the habit of no habit is the worst habit in the world. (Wales)
  • What is bad luck for one is good luck for another. (Ghana)
  • Good luck is the guardian of the stupid. (Sweden)
  • A change is as good as a rest. (England)
  • Good scribes are not those who write well, but who erase well. (Russia)
  • There is no such thing as a pretty good omelette. (France)
  • Of all the thirty-six alternatives, running away is the best. (China)


Three ancient problems are famously impossible to solve using a compass and straightedge alone: doubling the cube, trisecting an angle, and squaring the circle. Surprisingly, the first two of these can be solved using origami.

In the first, doubling the cube, we’re given the edge of one cube and asked to find the edge of a second cube whose volume is twice that of the first; if the first cube’s edge length is 1, then we’re trying to find \sqrt[3]{2}. Begin by folding a square of paper into three equal panels (here’s how). Then draw up bottom corner P as shown above, so that it’s touching the top edge while the bottom of the first crease, Q, touches the second crease as shown. Now point P divides the top edge into two segments whose proportions are 1 and \sqrt[3]{2}.

To trisect an angle, begin by marking the angle in one corner of a square (here’s it’s CAB). Make a horizontal fold, PP’, anywhere across the square. Then divide the space below this crease in half with another crease, QQ’. Fold the bottom left corner up so that corner A touches QQ’ (at A’) and P touches AC. Now A’AB is one-third of the original angle, CAB.

The first of these constructions is due to Peter Messer, the second to Hisashi Abe. Strictly speaking, each uses creases to produce a marked straightedge, which is not allowed in classical construction, but they’re pleasingly simple solutions to these vexing problems. There’s more at origami wizard Robert Lang’s website.

Cruel and Unusual

A king is angry at two mathematicians, so he decrees the following punishment. The mathematicians will be imprisoned in towers at opposite ends of the kingdom. Each morning, a guard at each tower will flip a coin and show the result to his prisoner. Each prisoner must then guess the result of the coin flip at the other tower. If at least one of the two guesses is correct, they will live another day. But as soon as both guesses are incorrect, they will be executed.

On the way out of the throne room, the mathematicians manage to confer briefly, and they come up with a plan that will spare them indefinitely. What is it?

Click for solution …

Page 1 of 87912345...10203040...Last »