(Bloomberg Opinion) -- Some puzzles are just puzzles. Others tell stories – often outlandish ones. What's a hint? What's just a red herring? It’s up to the solver to sort that out – hopefully while having a lot of fun.
This week's Conundrum is one such enigma, dropping prospective solvers smack in the middle of a somewhat fantastical tale:
Sam had just finished preparations for his socially-distanced dinner party, including Instagram-ready placecards made from letters painstakingly cut out of surprisingly durable gold foil.
But when Sam took a quick break to smoke his pipe, his cat ran across the table. Now everything is scrambled up. On top of that, Sam can’t shake the feeling that he’s left someone out. Can you help him figure out who that is?
DFGNLA
EYRM
SLOEGL
IMIL
DFOR
IOMBRO
PIPIP
If you uncover the answer -- or even make partial progress -- please let me know at skpuzzles@bloomberg.net before midnight Eastern time on Wednesday, June 3. (If you get stuck, there’ll be a hint announced in Bloomberg Opinion Today on Tuesday, June 2. Sign up here.)
Last Week’s Conundrum
We faced a hefty challenge: searching 20 stacks of 20 golden coins for a counterfeit stack – but with only enough battery power in our scale for three weighings.
Twenty stacks is far more than we could weigh individually.
But it turns out that if you know how much individual coins and counterfeits weigh, then you actually only need one weighing to find the fakes: If you take one coin from the first stack, two from the second, three from the third, and so forth, then you end up with 210 coins -- and if they were all real, then the total weight would be 210 times the weight of a real coin.
With a counterfeit stack, the total weight will be off by some amount depending on how many counterfeit coins ended up in the pile. And we can use that “error” to figure out which stack holds the fakes.
For example, if real coins weigh 10 grams, then a pile of 210 real coins would weigh 2100. If counterfeit coins weigh 15 – that is, 5 grams more than a real coin – then our pile will weigh an additional 5 grams for each counterfeit. So if our pile were to weigh in at 2110 grams, then we’d know it has to contain exactly two counterfeits – and because of the way we constructed the pile, those must have come from the second stack.
Luckily, it turns out that you can figure out the weights with only two weighings. There are actually two different approaches that work here:
- Weigh a single coin from some stack, and then weigh one coin from each stack together. Most likely, the 20 coins weigh a bit more than 20 times the weight of the single coin; in that case, you know the single coin was genuine, and can compute the difference in weights between real and counterfeit. (Of course, there’s a small chance the first coin you weigh is counterfeit, in which case the 20 coins will weigh much less than you would have expected. But that’s no problem. In this case, you get lucky and find the counterfeits with only two weighings!)
- Split the stacks into two sets of 10 and weigh each set. One of these sets will be heavier than the other; that set must contain the counterfeit stack. Knowing the weight of the other set lets you compute the weight of a real coin (it’s just the weight of the set divided by 200). With that, you can work out the weight of a counterfeit coin as well. (This strategy has the additional advantage that it cuts down the number of stacks you have to search in the third weighing -- you only have to check the 10 stacks in the heavier set, which means you can use the strategy above with a pile of 55 coins instead of 210.)
Combining either approach with the pile strategy lets us find the counterfeits faster than even Frank Abagnale could. Not too hard a lift after all! Plus, this solution works no matter how many stacks you have, so long as each stack has as many coins as there are stacks.
Benjamin Phillabaum was first to solve this Conundrum, just hours after it hit the web. Others among the 52 solvers included Anna Collins, Jack DeStories, Joshua Fenttiman, Joaquin Haces-Garcia, Emilio Grisolia, David Hollander, Sarvesha K, Dianhui Ke, Soebagio Notosoehardjo, and Lee Wilson. Family solving rivalries continue: This time, Duncan Rheingans-Yoo beat his brother Ross to the answer by over a day.
The team at Advantage Trading figured out a more complex solution that doesn’t rely on knowing whether counterfeits are heavier or lighter than real coins. And Robin Houston was able to show that you can still find the counterfeits even given 64 stacks of 20 coins each; it’s possible that’s as far as you can go.
The Bonus Round
Drop whatever you’re doing right now and watch squirrel ninja warrior (hat tip: Elizabeth Sibert). After that, hone your software engineering skills by defeating an ogre; go meet J.K. Rowling’s Ickabog; or just build a giant kookaburra in your yard (hat tip: Zoe DeStories). Send emails straight to spam; explore cryptograffiti and secret codes in Congressional clocks; or meet beings that feed on bitcoin. And inquiring minds want to know: what’s up with eels, anyway?
That's just over 4.5 pounds – substantially less than the weight of $15,000 in dollar coins.
Suhayl Chettih pointed out that the “golden” coins probably aren’t made of real gold, or else it would be hard to find a counterfeit that looks similar but is heavier. (Osmium, while far denser than gold, is bluish-white.)
That said, you need some serious skills to keep track of which coins came from which stacks when you’re doing your weighing.
Thanks also to Noam Elkies and Daniel Kane, who play-tested the puzzle in advance.
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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