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Kominers’s Conundrums: Secrets of a Magical Number

Scott Duke Kominers
·6 mins read

(Bloomberg Opinion) -- This is a very, very special edition of Conundrums. Why’s that, you ask? It’s the twenty-fourth edition. And 24 is a magical number.

First off, 24 is a factorial. That is, it’s the product of all the positive integers less than or equal to some number – in this case 4. Factorials are so cool that mathematicians typically denote them with an exclamation point (4! = 4 × 3 × 2 × 1 = 24). And they’re rare. Unless we significantly speed up our publication cycle, the next factorial-numbered Conundrum won’t be for years.

The incredible 24 is also a Tribonacci number, the number of faces in a Tesseract, and the largest number such that subtracting 1 from any of its divisors (other than 1 and 2) yields a prime.(1)

There are so many ways to construct 24 out of other numbers that inventor Robert Sun made a game out of it. In “24“ (the card game, not the television series), you’re given sets of four integers, and the goal is to form 24 from them using just the standard mathematical operations of addition, subtraction, multiplication and division (with the freedom to add parentheses wherever you want).

You have to use each number exactly once, so 4 × (6 – 1) + 4 = 24 is a valid solution for the set of integers 1, 4, 4, 6 – but just 4 × 6 isn’t. And some sets of integers are much harder to do this with than others. 2, 3, 5, 12, for example, turns out to be especially difficult.

For this week’s Conundrum, we put together six 24 puzzles for your solving pleasure. That’s right: 24 integers, divided into sets of four, and your goal is to make 24 from each set.

2, 3, 8, 8 1, 3, 4, 6 3, 4, 9, 10 4, 4, 10, 10 2, 2, 2, 64 1, 2, 5, 24

Because we support all kinds of math at Conundrums, you shouldn’t feel limited to just the basic operations. Any mathematical operation is fair game – logarithms, roots, you name it! We’ll be giving extra points for creative solutions. Just make sure that if an operation requires a number as input, you’re using one of the numbers from the puzzle. So you can only take a square root if you have a 2 available, for example.

That’s the full Conundrum, but once you’re done, there are two bonus challenges:

First, can you combine all 24 of the integers above to make 24 once more? And then take a crack at this one that my editor said was too hard: 2, 13, 15, 72.

If you Jack Bauer your way out of this seemingly impossible mission – or if you even make partial progress – please let us know at skpuzzles@bloomberg.net before midnight New York time on Thursday, October 1. (Special kudos to anyone who solves within 24 hours!)

If you get stuck, there’ll be hints announced on Twitter and in Bloomberg Opinion Today.) To be counted in the solver list, please include your full name with your answer.

Programming note: Next week, Conundrums will run on Sunday, October 4 at 8 a.m. New York time. If you have opinions about the optimal release day/time for the column, please let us know at skpuzzles@bloomberg.net.

Previously in Kominers’s Conundrums …

To commemorate the Ig Nobel Prizes, we presented topsy-turvy descriptions of 14 Nobel laureates’ prize-winning work. Once you figured out who all of them were, sorting by the year they won the Nobel would spell out the “sweet success none of them quite got the chance to taste:” NOBEL PEACH PIES.(2) (And indeed, none of our laureates were Peace Prize winners.)

Nernst (1920) – “German CHEMIST whose third focus was an absolute zero.” [Third law of thermodynamics: constant entropy at absolute zero.] O’Neill (1936) – “AUTHOR of plays featuring long English speeches that journey into nightly tragedy.” [“Long Day’s Journey into Night”; use of American English vernacular speeches.] Bellow (1976) – “Everyman AUTHOR who could have been a Rain King instead.” [“Henderson the Rain King.”] Elytis (1979) – “AUTHOR who struggled through modern Greek poems.” [“Against the background of Greek tradition, [Elytis] depicts […] modern man’s struggle for freedom and creativeness.”] Lee (1986) – “Taiwanese CHEMIST who crossed beams, but intentionally.” [Crossed molecular beam technique for measuring dynamics of chemical processes.] Pedersen (1987) – “Organic CHEMIST who discovered one ring to bind them.” [Crown ethers, which bind cations.(3)] Elion (1988) – “Pioneering female pharmacologist who used a backwards method to make MEDICINE.” [Rational drug design.] Altman (1989) – “Canadian-American CHEMICAL biologist who figured out that a certain protein wasn’t as catalytic as we might have thought.” [Catalytic properties of RNA (rather than an associated protein).] Corey (1990) – “Organic CHEMIST who used retro synthesizers.” [Retrosynthetic analysis.] Heaney (1995) – “Irish AUTHOR of poems about frogspawn and peat bogs who also re-transcribed old adventure literature.” [“Death of a Naturalist”; “Beowulf.”] Prusiner (1997) – “MEDICAL neurologist who figured out why some cows go ‘mad.’” [Prions.] Ignarro (1998) – “American MEDICAL pharmacologist who more or less discovered Viagra.” [Signaling properties of nitric oxide.] Ertl (2007) – “German physical CHEMIST who poured things on surfaces and watched them move around.” [Surface chemistry.] Shapley (2012) – “Mathematical ECONOMIST who proposed algorithm for achieving divorce-free marriage, which has never been used for that purpose.” [Theory of stable matching.]

Zoz solved first, followed by Sanandan Swaminathan, Elizabeth Sibert, Skylar Sukapornchai, Michael Branicky, and Suproteem Sarkar. FiveThirtyEight’s Riddler, Zach Wissner-Gross, submitted a sequence of four emoji: a prize, a peach, a slice of pie, and a LOL.

The Bonus Round

Call Juneau’s joke hotline; try this stack-folding challenge; or just reprogram your old Game Boy by playing it. Life-size Gundams; crows learning physics; hacking the “random-number bible” (hat tip: David Garbasz); and winning too many Chuck E. Cheese tickets. Play Minecraft as a fish and/or bring your hand-colored sea creatures to life. Old-school Disney sound effects. And inquiring minds want to know: Why do Washington and Baltimore look different from space?

(1) 24 also figures into many beautiful questions and structures in higher-dimensional geometry, including one I worked on back when I was mostly a number theorist.

(2) Thanks to the Animaniacs for teaching me this pun years ago.

(3) Note the Lord of the Rings reference pointing to a ruler.

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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