The Return of Monty Hall

In 2003, Danish computer scientist Peter Bro Miltersen discussed a surprisingly effective technique by which a player might guess the colors of slips of paper hidden in boxes (PDF). As this circulated in the mathematical community it evolved into a puzzle in which a group of 100 prisoners must find their own names on slips of paper. I wrote about it in 2011.

When Eugene Curtin and Max Warshauer wrote about the prisoner puzzle in The Mathematical Intelligencer in December 2006, reader A.S. Landsberg offered a variant called “The Return of Monty Hall.” On a new game show for couples, there are three curtains, which hide a key, a car, and a goat. One member of the couple is the “car-master” — she must find the car. The other is the “key-master” — he must find the key. If both succeed in their tasks, they win the new car. If either fails, they win the goat.

The key-master is led out of the room, where he can’t observe the proceedings, and then the car-master has two tries to find the car (open one curtain, and if the car isn’t there, open another curtain). If she finds the car, then all the curtains are closed again and the key-master is brought on to find the key. No communication at all is permitted between the two at this point. As before, the key-master has two tries to find the key by opening curtains.

If the couple play optimally, their odds of winning the car are a surprising 2/3. They do this using Miltersen’s technique. The car-master is Player #1, the key-master is Player #2, the car is Prize #1, the key is Prize #2, and the goat is Prize #3. The strategy is simply for each player to start by opening the curtain corresponding to his or her own player number, and if unsuccessful to open the curtain number corresponding to the prize number that the first curtain reveals. So, for example, the car-master, who is Player #1, begins by opening Curtain #1. If she finds the car then she’s done; if she finds the key (Prize #2) then she opens Curtain #2, and if she finds the goat (Prize #3) then she opens Curtain #3. When the curtains are reclosed, the key-master begins his turn by opening Curtain #2 (since he’s Player #2) and following the same plan.

That’s it. It’s not guaranteed to work, but it’s a simple strategy that requires minimal preparation and no communication at all once the game has begun. The universe of possibilities is so small that we can simply count them — here are the various arrangements of prizes and the resulting outcomes:

car-key-goat: win
car-goat-key: win
key-goat-car: lose
key-car-goat: win
goat-key-car: win
goat-car-key: lose

Landsberg’s letter brought a comment by reader Eric Grunwald, who pointed out that a third person can be introduced to the Monty Hall game without reducing the overall chance of success. Replace the goat with a GPS system and add a third contestant, the “GPS-master.” Following the same rules, and again forbidding any communication among the contestants, Miltersen’s strategy ensures a 2/3 probability that all three players find their prizes.


  • Consecutive U.S. presidents Grant, Hayes, and Garfield were all born in Ohio and served as Civil War generals.
  • Travel due south from Buffalo and you’ll reach the Pacific Ocean.
  • Oliver Wendell Holmes Jr. shook hands with both John Quincy Adams and John F. Kennedy.
  • This false statement is not self-referential.
  • “When you have no basis for an argument, abuse the plaintiff.” — Cicero

In the 2004 film Shark Tale, the shark Lenny coughs up several items onto a table. Among them is a Louisiana license plate, number 007 0 981. The same plate is retrieved from sharks in both Jaws and Deep Blue Sea.


Where does power come from? To be legitimate, a law must be enacted by a suitably constituted authority. But this authority must be constituted by some previously enacted law. This chain can’t continue backward forever; there must be some highest authority that appeals to a basic norm rather than to a foregoing set of rules. But how can the legal existence of this basic norm be established? There seem to be only two possibilities:

  1. The basic norm is enacted law. Since it’s not enacted by any other authority, this means it’s enacted by the highest authority itself.
  2. The basic norm isn’t enacted law. This means that its validity isn’t derived from that of any other norm but is an “original fact” that’s needed to underwrite the validity of every other norm in the system.

“Two, and only two, answers seem possible,” writes University of Copenhagen philosopher Alf Ross. “But both seem unacceptable. That is the puzzle.”

Somewhat related: The United Kingdom’s High Court of Chivalry was created in the 14th century to consider cases of the misuse of heraldic arms. It had been silent for centuries when suddenly in 1954 it was called on to hear a case: The Palace Theatre was displaying the arms of the Manchester city council on its seal, suggesting a link between the two. Before hearing the case, the court first had to rule on whether it still existed. It decided that it did. (And the city council won.)

(Alf Ross, “On Self-Reference and a Puzzle in Constitutional Law,” Mind 78:309 [January 1969], 457-480.) (Thanks, Julian.)


‘As to moral courage, I have very rarely met with the two o’clock in the morning courage. I mean, unprepared courage, that which is necessary on an unexpected occasion, and which, in spite of the most unforeseen events, leaves full freedom of judgment and decision.’

— Napoleon, to Emmanuel, comte de Las Cases, Journal of the Private Life and Conversations of the Emperor Napoleon at Saint Helena, 1824

Podcast Episode 109: Trapped in a Cave
Image: Wikimedia Commons

In 1925, Kentucky caver Floyd Collins was exploring a new tunnel when a falling rock caught his foot, trapping him 55 feet underground. In this week’s episode of the Futility Closet podcast we’ll follow the desperate efforts to free Collins, whose plight became one of the first popular media sensations of the 20th century.

We’ll also learn how Ronald Reagan invented a baseball record and puzzle over a fatal breakfast.

Sources for our feature on Floyd Collins:

Robert K. Murray and Roger W. Brucker, Trapped!, 1979.

Gary Alan Fine and Ryan D. White, “Creating Collective Attention in the Public Domain: Human Interest Narratives and the Rescue of Floyd Collins,” Social Forces 81:1 (September 2002), 57-85.

“Floyd Collins Is Found Dead,” Madison Lake [Minn.] Times, Feb. 19, 1925.

Associated Press, “Sand Cave Is to Be Grave of Explorer,” Feb. 18, 1925.

Associated Press, “Floyd Collins Will Be Left in Sand Cave for His Last Sleep,” Feb. 18, 1925.

Associated Press, “Ancient ‘Floyd Collins’ Found in Mammoth Cave,” June 19, 1935.

Ray Glenn, “Floyd Collins Trapped in Cave 35 Years Ago,” Park City [Ken.] Daily News, Feb. 7, 1960.

Carl C. Craft, “Floyd Collins Case Recalled After 40 Years,” Kentucky New Era, Feb. 1, 1965.

William Burke Miller, “40 Years Ago, World Prayed for Floyd Collins,” Eugene [Ore.] Register-Guard, Feb. 11, 1965.

Paul Raupp, “Floyd Collins Finds Final Resting Place,” Bowling Green [Ken.] Daily News, March 26, 1989.

Listener mail:

Howard Breuer et al., “Dumb Criminals,” People 81:1 (Jan. 13, 2014).

Ronald Reagan, “Remarks at a White House Luncheon for Members of the Baseball Hall of Fame, March 27, 1981,” The American Presidency Project.

Ronald Reagan, An American Life, 1990.

This week’s lateral thinking puzzle was contributed by listener Stephen Harvey.

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 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 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!

Distant Early Warning

Alexandre Dumas’ cat knew when he was coming home:

At the time I speak of, I held a situation in the service of the Duc d’Orléans, with a salary of 1500 francs. My work occupied me from ten in the morning until five in the afternoon. We had a cat in those days, whose name was Mysouff. This cat had missed his vocation; he ought to have been a dog. Every morning I started for my office at half-past nine, and came back every evening at half-past five. Every morning Mysouff followed me to the corner of a particular street, and every evening I found him in the same street, at the same corner, waiting for me. Now the curious thing was that on the days when I had found some amusement elsewhere, and was not coming home to dinner, it was of no use to open the door for Mysouff to go and meet me. Mysouff, in the attitude of the serpent with its tail in its mouth, refused to stir from his cushion. On the other hand, on the days I did come, Mysouff would scratch at the door until some one opened it for him.

“My mother was very fond of Mysouff,” he wrote. “She used to call him her barometer.”

Coming and Going

In 1978 the Chronicle of Higher Education mentioned an old exam question:

Q. How far can a dog run into the woods?

A. Halfway. The rest of the time he is running out.

Harvard’s Richard E. Baym wrote in to take issue with the answer:

The correct answer is ‘All the way’. Certainly we understand that the dog is running ‘in’ only until he reaches the middle of the forest, but this is in fact, all the way in. If the dog ran only half ‘in’, he would not yet be at the middle. Indeed if the dog ran halfway in and then ran halfway out, he would still be in the woods.

The editors noted, “It occurs to us that the dog’s continued presence there would be useful, in case something happens to that tree that we’ve been hearing about since high school physics — the one that falls when no one is in the forest and since there is no eardum to register sound waves, makes no noise. You know what a fine sense of hearing a dog has. Let him run halfway in (or as Mr. Baym argues, all the way), settle there, and keep an ear cocked for that tree.”

(from Robert L. Weber, ed., Science With a Smile, 1992.)


In presenting the rules of chess, some writers carelessly say that a pawn that reaches the eighth rank can be promoted to any piece that the player chooses. That’s a bit too generous, as a couple of puzzle composers have noted. In 1941 Leonid Kubbel presented this problem — White is to mate in two moves:

kubbel promotion puzzle

It’s not immediately clear how to release Black from his stalemate and still mate him on the next move. The solution is to promote the e7 pawn to a black king!

kubbel promotion puzzle - solution

Now it’s Black’s move — he has to play 1. … Kd8, and White can mate both kings with 2. Qd7#!

The Polish master Johannes Zukertort offered this one: White is to mate on the move:

zukertort promotion puzzle

Here White promotes the pawn to a black knight, ending the game. (Note that it must be a knight — crazy as it seems, this is the only black piece that produces mate.)

Divide and Conquer

Facing dental surgery one day, mathematician Matt Parker asked Twitter for a math puzzle to distract him. A friend challenged him to put the digits 1-9 in order so that the first two digits formed a number that was a multiple of 2, the first three digits were a multiple of 3, and so on.

Leaving the digits in the conventional order 1234356789 doesn’t work: 12 is divisible by 2 and 123 by 3, but 1234 isn’t evenly divisible by 4. “By the end of my dental procedure, I had some but not all of the digits worked out, but, apparently, you’re not allowed to stay in the dentist’s chair after they’re finished.” At home he finished working out the solution, which is unique. What is it?

Click for Answer

Last Words

Note found in the pocketbook of an English corporal killed during the battle of the Somme:

Dear Mother

I am writing these few lines severely wounded. We have done well our Batt. advanced about 3 quarters of a mile. I am laid in a shell hole with 2 wounds in my hip and through my back. I cannot move or crawl. I have been here for 24 hours and never seen a living soul. I hope you will receive these few lines as I don’t expect anyone will come to take me away, but you know I have done my duty out here now for 1 year and 8 months and you will always have the consolation that I died quite happy doing my duty.

Must give my best of love to all the cousins who [have] been so kind to me since I have been out here and the Best of love to Arthur and Harry and all at Swinefleet. xxx

He was identified as John Duesbery of Bradford, and the note and his other belongings were sent home to his family. In The Quick and the Dead, Richard van Emden writes, “His grave would have been marked in a rudimentary way, perhaps with a piece of wood or an upturned rifle, but whatever was placed there it was subsequently destroyed and John’s body lost.”