The Pirate’s Solution

Hello all,

Last week we posted a Long-John-Silver-ized version of a classic puzzle. Here’s how it went:

Captain Montgomery Hall, the famous pirate, was getting ready to retire, and wanted to be sure he had a qualified successor, so one morning he came on deck with three buckets, their tops covered. As the crew watched he hoisted each of them overhead on separate halyards.

“To be a captain,” he said, “Ye have to be smart. Who amongst ye is the smartest?”

The pirates conferred a bit, and then pushed forward their reluctant choice.

“All right, we’ll see how smart ye are. There’s a gold doubloon in one a’them buckets. There’s a piece of stale, weevily bread in the other two. Pick the right bucket, and you get the doubloon, and the command. Pick the wrong one, ye get yer usual breakfast.”

“But captain,” said the sailor, “No matter how smart I might be, I could only pick the right bucket by luck.”

“Just by sayin’ that, it shows that at least ye aren’t stupid. But pick one anyway.”

So the sailor chose one of the buckets, walked over to the pin its halyard was on, and prepared to cast off.

“Wait just a minute,” said the captain, “Let’s make this more interesting.” And he went over to one of the other pins, lowered one of the other buckets to deck, and removed its cover. Inside was a piece of stale, weevily bread.

“This means that the doubloon is either in the bucket ye chose, or in the third bucket,” said the captain. Could be either one, right?”

“Ye-e-ss,” said the sailor thoughtfully.

“I’ll let ye change your mind, if ye want,” said the captain.

After the sailor considered the matter for a while, he walked away from his first choice, and lowered the third bucket to deck.

So here is the puzzle: Was this the smart thing to do? By changing his mind, did the sailor improve his chances, or make them worse, or did they stay the same?

The solution is tricky, which is why, for this puzzle, any answer right or wrong qualified entrants for the prize drawing. And the winner of that drawing is…. my dear old friend Jim Cole who, along with his spouse Barbara have been cruising the South Pacific recently. Jim is an actual rocket scientist, but even his answer wasn’t quite correct. But as promised, here is my utterly subjective choice for best explanation, courtesy of Jerry Andrews, who wrote:

“The best choice (by a factor of 2) is to pick the OTHER bucket. When the pirate picks a bucket, he has a 1 in 3 chance o’ bein’ right, and a 2 in 3 chance o’ bein’ wrong. Now the captain knows which bucket bears the doubloon, so he’ll always lower a bucket with weevily bread—he’ll never choose the bucket with the doubloon, whether the pirate picked it not. Since the captain’s choice isn’t random, the odds that the pirate is wrong haven’t changed—they’re still 2 in 3. By changing his choice, he’s doubled his chance o’ bein’ right, and becomin’ captain himself. By stickin’ with his original choice, he [would’ve] proved he’s a regular pirate himself, and may not be bright enough to be captain. On the other hand, it’s usually better to be lucky than good, so maybe he’ll win the job anyway.”

I liked this answer because it was about as concise as possible, as well as being expressed in piratical patois.  Jerry, Jim, you are hereby awarded a genuine Rigger’s Apprentice T-shirt. Let me know your sizes and mailing addresses and we will get them shipped.

By the way, the first version of this puzzle was published by Joseph Bertrand in 1889, and people have been arguing about it ever since. It is best-known today as the Monty Hall Puzzle (hence the name of the pirate captain). There is a considerable body of commentary around the puzzle, with my favorite being that spurred by Marilyn Vos Savant’s version in Parade magazine, back in 1990. You can read one account of this here: http://www.stayorswitch.com/history.html.

The problem is that the correct answer is so deeply counterintuitive that most non-experts — and even many experts

— refuse to believe the answer, even when it has been carefully explained.

We’ll be posting a helm balance puzzle in a week, so be sure to check in. If you would like to be notified when we post a puzzle or article, you can subscribe at the bottom of this post, or any of my other posts.