Dividing the Spoils


Ten pirates have 100 gold pieces and want to divide them according to the law of the sea, which says that the spoils go to the strongest. So they arrange themselves from weakest to strongest, P1, P2, …, P10. But these are democratic pirates, so they ask the strongest pirate to make a proposal as to how to divide the loot. All 10 pirates will then vote on it. If at least 50 percent of them support the proposal, then they’ll enact it and that’s that. Otherwise the proposer will be thrown to the sharks.

All pirates value their lives more than gold, all are rational, they cannot cut the gold pieces into smaller pieces, and no pirate will agree to a side bargain to share pieces. What proposal should the strongest pirate make in order to get the most gold?

Click for Answer

“A Postal Problem”


Browsing the Post Office Guide in June 1891, Lewis Carroll discovered an ambiguity that produces “a very curious verbal puzzle” — he sent this pamphlet to friends and interested parties:

The Rule, for Commissions chargeable on overdue Postal Orders, is given in the ‘Post Office Guide’ in these words, (it is here divided, for convenience of reference, into 3 clauses)—

(a) After the expiration of 3 months from the last day of the month of issue, a Postal Order will be payable only on payment of a Commission, equal to the amount of the original poundage;

(b) with the addition (if more than 3 months have elapsed since the said expiration) of the amount of the original poundage for every further period of 3 months which has so elapsed;

(c) and for every portion of any such period of 3 months over and above every complete period.

You are requested to answer the following questions, in reference to a Postal Order for 10/- (on which the ‘original poundage’ would be 1d.) issued during the month of January, so that the 1st ‘period’ would consist of the months February, March, April; the 2nd would consist of the months May, June, July; and the 3rd would consist of the months August, September, October.

(1) Supposing the Rule to consist of clause (a) only, on what day would a ‘Commission’ begin to be chargeable?

(2) What would be its amount?

(3) Supposing the Rule to consist of clauses (a) and (b), on what day would the lowest ‘Commission’ begin to be chargeable?

(4) What would be its amount?

(5) On what day would a larger ‘Commission’ (being the sum of 2 ‘Commissions’) begin to be chargeable?

(6) What would be its amount?

(7) On what day would a yet larger ‘Commission’ begin to be chargeable?

(8) What would be its amount?

(9) Taking the Rule as consisting of all 3 clauses, in which of the above-named 3 ‘periods’ does clause (c) first begin to take effect?

(10) Which day, of any ‘period,’ is the earliest on which it can be said that a ‘portion’ of the ‘period’ has elapsed?

(11) On what day would the lowest ‘Commission’ begin to be chargeable?

(12) What would be its amount?

(13) On what day would a larger ‘Commission’ begin to be chargeable?

(14) What would be its amount?

(15) On what day would a yet larger ‘Commission’ begin to be chargeable?

(16) What would be its amount?



He followed up with this supplement later that month:


The trouble, as I read it, is that clause (c) is ambiguous. Presumably the postal authorities intended the general rule to be that a patron had three months to redeem a postal order, and beyond this would be charged a commission (here, 1 penny) for every three months that had elapsed since the deadline — including the last such period, which would not be prorated. Unfortunately, the phrase “every complete period” means exactly that — it refers to every completed period on the calendar. This sets the clock going twice as fast as intended. Our patron should owe 1d on May 1, 2d on August 1, and 3d on November 1. But with clause (c) worded as it is, she’ll owe 1d on May 1, 4d on August 1, and 6d on November 1. The final effect is that, beyond the first period, postal patrons who follow these rules will pay twice the intended commission.

I don’t know whether the post office ever learned about this. I imagine most patrons trusted them to do the math, and no one but Carroll recognized the ambiguity.

The Shortest Road

shortest road - 1

We want to build a road between two cities, A and B, that are separated by a river. We can build a bridge, but it must be perpendicular to the river’s banks, as shown. Where along the river’s length should we place the bridge if we want to minimize the total length of the road?

Click for Answer

Podcast Episode 128: The Battle for Castle Itter

Image: Wikimedia Commons

The closing days of World War II witnessed a bizarre battle with some unlikely allies: American and German soldiers joined forces to rescue a group of French prisoners from a medieval castle in the Austrian Alps. In this week’s episode of the Futility Closet podcast we’ll follow the Battle for Castle Itter, the only time that Allies and Germans fought together in the war.

We’ll also dodge another raft of aerial bombs and puzzle over a bottled pear.


In 1917, Royal Flying Corps trainee Graham Donald fell out of his plane at the top of a loop.

In 1750, the 1st Earl of Hardwicke installed an artificial ruin near his country house, Wimpole Hall.

Sources for our feature on the Battle for Castle Itter:

Stephen Harding, The Last Battle, 2013.

Stephen Harding, “The Battle for Castle Itter,” World War II 23:3 (August/September 2008), 38-45.

George Hodge, “The Last Battle: When U.S. and German Soldiers Joined Forces in the Waning Hours of World War II in Europe,” Military Review 94:4 (July/August 2014), 100.

John G. Mayer, “12th Men Free French Big-Wigs,” 12th Armored Division Hellcat News, May 26, 1945.

Andrew Roberts, “World War II’s Strangest Battle: When Americans and Germans Fought Together,” Daily Beast, May 12, 2013.

Bethany Bell, “The Austrian Castle Where Nazis Lost to German-US Force,” BBC News, May 7, 2015.

Listener mail:

Roadside America, “Omaha, Nebraska: Plaque: Japanese Balloon Bomb Exploded Here,”.

“B-52 Accidentally Bombs Kansas Lake,” Aero News Network, Dec. 16, 2006.

Bill Kaczor, “Bombs Rained on Florida Family in 1944,” Los Angeles Times, Aug. 14, 1994.

Wikipedia, “MOVE: 1985 bombing” (accessed Nov. 4, 2016).

Wikipedia, “Pavlovsk Experimental Station” (accessed Nov. 4, 2016).

Ian Crofton, A Curious History of Food and Drink, 2014.

Wikipedia, “1958 Tybee Island Mid-Air Collision” (accessed Nov. 4, 2016).

This week’s lateral thinking puzzles were adapted from the Soviet popular science magazine Kvant and the 2000 book Lateral Mindtrap Puzzles and contributed by listener Steve Scheuermann. We refer to this image in the second puzzle:

Image: Wikimedia Commons

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!

Animal Spirits

Hatebeak (above) is a death metal band fronted by a Congo African grey parrot named Waldo. To date they’ve released four albums, Beak of Putrefaction, Bird Seeds of Vengeance, The Thing That Should Not Beak, and Number of the Beak.

Bird Seeds of Vengeance was made with Caninus, a deathgrind band led by two pitbull terriers, Basil and Budgie. Unfortunately Basil died in 2011, and founding member Budgie died earlier this year, so the band has now retired.

“We are all lucky to have had her in our orbit for as long as we did,” the surviving members wrote after Budgie’s death. “She touched many lives, licked many faces, pushed many people out of beds, stole many slices of pizza, ate many soundguy burritos and, most importantly, inspired many to adopt from shelters instead of buying from pet shops or online breeders.”

Black and White

dawson chess puzzle

From T.R. Dawson, a logic problem posing as a chess puzzle. If pinned men do not check, how can White mate in two moves?

Click for Answer

Ups and Downs


Here’s how the Union enciphered its messages during the Civil War. Abraham Lincoln sent this dispatch on June 1, 1863:


The first word, GUARD, indicates the size of a containing rectangle and the paths on which the words must be laid out to decipher the message. In this case, they’ll go up the first column, down the second, up the fifth, down the fourth, and up the third. Also, just to confuse the Confederates, every eighth word after GUARD is a null and should be discarded. So we get:

WAYLAND AT             ODOR     ARE        DETAINED
THEY    ARE            DETAINED AND        GET
THEM    OFF            IF       YOU        CAN
ADAM    NELLY          THIS     FILLS      UP

The last steps are to remove THIS FILLS UP, which is only there to fill out the block, and to replace a few code words:

VENUS = colonel
WAYLAND = captured
ODOR = Vicksburg
NEPTUNE = Richmond
ADAM = President of the United States
NELLY = 4:30 p.m.

That gives us the final message:

For Colonel Ludlow,

Richardson and Brown, correspondents of the Tribune, captured at Vicksburg, are detained at Richmond. Please ascertain why they are detained and get them off if you can.

The President, 4:30 p.m.

This system was such a valuable source of breaking news that Lincoln often visited the military telegraph office in the War Department, next to the White House, and would chat with the operators there. One of them, David Homer Bates, who was only 18 when the war started, remembered, “Outside the members of his cabinet and his private secretaries, none were brought into closer or more confidential relations with Lincoln than the cipher-operators, … for during the Civil War the President spent more of his waking hours in the War Department telegraph office than in any other place, except the White House. … His tall, homely form could be seen crossing the well-shaded lawn between the White House and the War Department day after day with unvaried regularity.”

(From David Kahn, The Codebreakers, 1996.)