The Jeep Problem
Image: Wikipedia

An adventurer wants to explore a desert. He has a jeep that can carry up to 1 unit of fuel at any time and that will travel 1 unit of distance on 1 unit of fuel. As he travels he can leave any amount of the fuel that he’s carrying at any point, as a fuel dump to be picked up later. He starts from a fixed base at the edge of the desert, where there’s an unlimited supply of fuel. How far into the desert can he go if he wants to return safely to the base at the end of each trip?

Surprisingly, with some intelligent planning he can go as far as he likes. The diagram above shows how far he can get with 3 trips:

  1. On the first trip he departs the base with 1 unit of fuel. He drives 1/6 unit into the desert, leaves 2/3 units of fuel at a fuel dump, and returns to the base using the remaining 1/6 unit of fuel.
  2. On the second trip he leaves the base with 1 unit of fuel, drives 1/6 unit to the fuel dump, and draws 1/6 unit from the dump. Now he’s carrying 1 unit of fuel. Then he drives 1/4 unit farther into the desert and leaves 1/2 unit at a new dump. Now he has 1/4 unit of fuel remaining, which is just enough to reach the first fuel dump, where he collects another 1/6 unit of fuel and returns to base.
  3. On the third trip he drives 1/6 unit to reach the first fuel dump, where he tops up with 1/6 unit of fuel (leaving 1/6 unit remaining there). Then he drives 1/4 to the second dump, where he collects 1/4 unit of fuel (again topping up to 1 full unit). (1/4 unit now remains in the second fuel dump.) Now he can drive 1/2 unit distance into the desert before he has to return to the second fuel dump, where he collects the remaining 1/4 unit fuel, which enables him to reach the first fuel dump, where he collects the last 1/6 unit of fuel, which is just enough to get back to base.

So if 3 trips are planned the explorer can travel a round-trip distance of 1 + 1/2 + 1/3 = 11/6 units. You can see the pattern: If the explorer had planned 4 trips, he would set up fuel dumps at distances of 1/8, 1/6, and 1/4 from the base, initially storing 3/4, 2/3, and 1/2 units at each and then drawing 1/8, 1/6, and 1/4 units of fuel from each on each visit. As before, on the final trip he could depart the last fuel dump with a full tank, drive 1/2 unit into the desert, and then return to the base, exhausting each fuel dump on the way. In that case he’d have traveled a round-trip distance of 1 + 1/2 + 1/3 + 1/4 = 25/12 units.

This is just the harmonic series, 1 + 1/2 + 1/3 + 1/4 + 1/5 + …, which is divergent — in principle, at least, the explorer can travel as far as he likes into the desert, provided he plans a large enough series of trips. In practice it would be very difficult, though — both the number of fuel dumps and the total amount of fuel necessary increase exponentially with the distance to be traveled.

Eternity in an Hour

At the end of his 1986 book Paradoxes in Probability Theory and Mathematical Statistics, statistician Gábor J. Székely offers a final paradox from his late professor Alfréd Rényi:

Since I started to deal with information theory I have often meditated upon the conciseness of poems; how can a single line of verse contain far more ‘information’ than a highly concise telegram of the same length. The surprising richness of meaning of literary works seems to be in contradiction with the laws of information theory. The key to this paradox is, I think, the notion of ‘resonance.’ The writer does not merely give us information, but also plays on the strings of the language with such virtuosity, that our mind, and even the subconscious self resonate. A poet can recall chains of ideas, emotions and memories with a well-turned word. In this sense, writing is magic.


graft puzzle

You’re a venal king who’s considering bribes from two different courtiers.

Courtier A gives you an infinite number of envelopes. The first envelope contains 1 dollar, the second contains 2 dollars, the third contains 3, and so on: The nth envelope contains n dollars.

Courtier B also gives you an infinite number of envelopes. The first envelope contains 2 dollars, the second contains 4 dollars, the third contains 6, and so on: The nth envelope contains 2n dollars.

Now, who’s been more generous? Courtier B argues that he’s given you twice as much as A — after all, for any n, B’s nth envelope contains twice as much money as A’s.

But Courtier A argues that he’s given you twice as much as B — A’s offerings include a gift of every integer size, but the odd dollar amounts are missing from B’s.

So who has given you more money?

Things and Stuff

Anchormen, chairs, dogs, flowers, and comets are things: If I have one anchorman and add another, I have two anchormen. My chair did not exist until it was assembled into that form. And if a comet hits Paraguay, it is no longer a comet.

Helium, gravy, wood, music, and joy are stuff: If some helium escapes my balloon, it seems wrong to say that I’ve lost a thing. If I divide my gravy into two portions, it’s still gravy. And if I chop my cabin into firewood, the amount of wood in the world does not seem to have changed.

We seem to distinguish between these two classes of existence. We can count things, but stuff forms a sort of cumulative mass. Things are made of stuff (crowns are made of gold), but stuff is made of things (gold is made of molecules). What’s at the bottom? And what leads us to make these distinctions?

(Kristie Miller, “Stuff,” American Philosophical Quarterly 46:1 [January 2009], 1-18.)


schnittke gravestone

Composer Alfred Schnittke’s gravestone bears a musical staff with a semibreve rest under a fermata, indicating that the rest should be held as long as desired. It’s marked fff, or fortississimo, meaning that it should be performed very strongly.

Overall it might be interpreted to mean “a decided rest of indefinite length.”


Why do we cultivate friendships? What reason do I have to be a friend to another person, that is, to care about him for his own sake? In order to make the friendship worthwhile, such a reason would have to explain how doing it makes my own life better. But that’s a problem: If I pursue the friendship in order to improve my own life, then I’m not really being a true friend, caring about my friend for his own sake.

University of Newcastle philosopher Joe Mintoff writes, “The problem is that, even though many of us think that being a true friend makes our lives better, paradoxically this thought had better not guide our pursuit of friendship, lest this mean that we are not true friends and that our lives are not made better.” Why, then, do we seek to befriend others?

(Joe Mintoff, “Could an Egoist Be a Friend?,” American Philosophical Quarterly 43:2 [April 2006], 101-118.)

The Two Errand Boys

dudeney errand boys

Another conundrum from Henry Dudeney’s Canterbury Puzzles:

A country baker sent off his boy with a message to the butcher in the next village, and at the same time the butcher sent his boy to the baker. One ran faster than the other, and they were seen to pass at a spot 720 yards from the baker’s shop. Each stopped ten minutes at his destination and then started on the return journey, when it was found that they passed each other at a spot 400 yards from the butcher’s. How far apart are the two tradesmen’s shops? Of course each boy went at a uniform pace throughout.

Click for Answer

Podcast Episode 134: The Christmas Truce

In December 1914 a remarkable thing happened on the Western Front: British and German soldiers stopped fighting and left their trenches to greet one another, exchange souvenirs, bury their dead, and sing carols in the spirit of the holiday season. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Christmas truce, which one participant called “one of the highlights of my life.”

We’ll also remember James Thurber’s Aunt Sarah and puzzle over an anachronistic twin.


In 1898, G.W. Roberts of Birmingham made a full-size piano from 3,776 matchboxes and 5 pounds of glue.

In 1892, 69 men raced 302 miles on stilts, from Bordeaux to Bayonne and Biarritz and back.

Sources for our feature on the Christmas truce:

Terri Blom Crocker, The Christmas Truce: Myth, Memory, and the First World War, 2016.

Stanley Weintraub, Silent Night: The Story of the World War I Christmas Truce, 2001.

Chris Baker, The Truce: The Day the War Stopped, 2014.

Peter Hart, “Christmas Truce,” Military History 31:5 (January 2015), 64-70.

Joe Perry, Christmas in Germany: A Cultural History, 2010.

Ian Herbert, “Muddy Truth of the Christmas Truce Game,” Independent, Dec. 24, 2014.

David Brown, “Remembering a Victory For Human Kindness,” Washington Post, Dec. 25, 2004.

“Alfred Anderson, 109, Last Man From ‘Christmas Truce’ of 1914,” New York Times, Nov. 22, 2005.

“The Christmas Truce, 1914,” The Henry Williamson Society (accessed Dec. 16, 2016).

Mike Dash, “The Story of the WWI Christmas Truce,” Smithsonian, Dec. 23, 2011.

Stephen Moss, “Truce in the Trenches Was Real, But Football Tales Are a Shot in the Dark,” Guardian, Dec. 16, 2014.

Listener mail:

Kirk Ross, The Sky Men: A Parachute Rifle Company’s Story of the Battle of the Bulge and the Jump Across the Rhine, 2004.

A short version of the barrel-of-bricks episode from MythBusters:

Listener Daniel Sterman recommends the original episode, “Barrel of Bricks,” from Oct. 10, 2003.

Wikipedia, “Sandman (Wesley Dodds)” (accessed Dec. 16, 2016).

Wikipedia, “Sala Gang” (accessed Dec. 16, 2016).

This week’s lateral thinking puzzle was suggested by listeners Greg Askins, Stacey Irvine, and Donald Mates. 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

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 Thanks for listening!