Kominers’s Conundrums: Pick a Number, Any Number at All

(Bloomberg Opinion) -- This is the 64th edition of Conundrums — a rare power of 2! We won’t see another of those for more than a year. And as is customary for the column, we’re celebrating this mathematically interesting edition with a math game.

The format is simple: Everyone submits a number between 0 and 64,000 (inclusive — you can submit 0 or 64,000 if you want). We’re going to take all of those numbers and compute their average. Your goal is for the number you submitted to be the closest to four-sixths of that average.

Got it? Whoever submits the number that is closest to four sixths of the average of all the numbers wins.

That’s all you have to do — guess a number!

It sounds easy, but it is deceptively subtle. You can’t just pick any number out of the blue — you have to think about how other people will approach the problem.

For example: If everyone is submitting numbers under 64,000, then the average has to be less than 64,000 — so four-sixths of that average must be less than 42,667. Since your goal is to hit four-sixths of the average, this means you certainly shouldn’t submit anything over 42,667, right?

But wait — everybody else can presumably make that same calculation, which means none of the numbers submitted should be over 42,667. But then the average has to be below 42,667, and four-sixths of that is 28,444! So maybe you shouldn’t submit anything higher than 28,444? It seems like the right number to send in just keeps getting smaller and smaller.

Of course, all of that is assuming that everyone is calculating carefully, and trying to win the game. It’s possible that some people will submit specific numbers just because they like them, or send in huge numbers to throw everyone else’s entries off.

This game has been played in newspapers and game theory classrooms all around the world — as well as in other puzzle columns. And one thing that’s been found is that different audiences play very different strategies.

So who are our Conundrums readers? We know they’re puzzle aficionados, but does that mean they’re rational game theorists? Will some people just submit their favorite numbers, irrespective of how likely those numbers are to win the contest? And if they do that, what could their favorite numbers be?

Consider carefully, and then send in your entry at the form here.

If you have any trouble with the form itself, please let us know at skpuzzles@bloomberg.net. We’ll be collecting entries until midnight, New York time on September 23. To be counted in the solver list, please include your name with your submission.

And don’t forget to sign up for our Conundrums email list!

Have You Solved our Last Conundrum? There’s Still Time!

Since we launched Conundrums, we’ve been presenting the solution to each puzzle in the subsequent column. But we put some extra space between editions in July and August, and heard from many people that they appreciated the extra solving time.

So we’re going to experiment with offsetting puzzles and solutions by more than one edition. That means you’ve got an extra week to solve our most recent escape room-style Conundrum. If you haven’t checked it out yet, you really should — it’s one of the most spectacular puzzles we’ve ever put together.

And if you have solved it already, you can continue the adventure with The Escape Game’s “The Heist” — the Conundrum is a prequel! Or you can just check out the bonus round below.

The Bonus Round

The Nobel nomination archives; and this year’s Ig Nobel prizes. Design your own escape room; or solve this upcoming puzzle contest from Labsterium. Earthquake pictographs (hat tip: Maya Ajmera); installing Windows 3.1; and stealing copper by spray painting rocks. “How an A.I. is Becoming the World’s Best Pokémon Player”; the weirdest “Super Mario” power-ups; and “The Most Dangerous Game” (hat tip: The ASAASA). The secret patterns of punctuation (hat tip: Clive Thompson); highly satisfying digital art; and heavy metal Billy Bass. “Pollock’s Fractals” (hat tip: Ellen Dickstein Kominers); and “The (Mathematical) Problem of Mondrian’s Paintings” (hat tip: Eddie Cheung). Plus inquiring minds want to know: Why can’t we make new Stradivari?

And yes, four sixths of the average is the same as two thirds of the average.

Some have even said that dinosaurs played it.

I see you, 5782 fans!

And one answer per person, please! We reserve the right to disqualify submissions that appear to be spammy.

This column does not necessarily reflect the opinion of the editorial board or Bloomberg LP and its owners.

Scott Duke Kominers is the MBA Class of 1960 Associate Professor of Business Administration at Harvard Business School, and a faculty affiliate of the Harvard Department of Economics. Previously, he was a junior fellow at the Harvard Society of Fellows and the inaugural research scholar at the Becker Friedman Institute for Research in Economics at the University of Chicago.

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