We all keep secrets. It’s easy to do once you’ve committed to it. Just don’t say anything. But it can be harder when you want to share a little bit of information, without your confidante inferring tea that you didn’t intend to spill. Computer scientists often draw this distinction between security and privacy. Security focuses on keeping sensitive data out of the hands of untrusted parties, full stop. Privacy, on the other hand, aims to give people fine-tuned control over how their personal data gets distributed and used. This kind of partial secrecy turns out to require some delicate mathematics beyond the demands of traditional security.
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In this week’s puzzle, you want to conceal your income from your coworkers, but you’re incentivized to share a little bit so that the group can calculate their average salary. I typically support wage transparency in the workplace, so thankfully the beautiful trick behind this puzzle extends well beyond cagey compensation conversations and enjoys wide-reaching applications in digital privacy.
Did you miss last week’s puzzle? Check it out here, and find its solution at the bottom of today’s article. Be careful not to read too far ahead if you haven’t solved last week’s yet!
Puzzle #23: Money Secrets
You and nine of your coworkers are sitting in a conference room when somebody wonders aloud about the group’s average salary. Everyone is curious, but nobody feels comfortable sharing their own pay with the group. How can you all learn your average salary without anybody learning any more information about another person’s salary (other than what can be deduced from the group average)?
You all have pen and paper and can conceal what you write from others, but you have no other tools at your disposal. You may assume that everybody cooperates with the chosen strategy.
This puzzle feels a little more open-ended than others in the series. I have a specific elegant solution in mind, but I’m eager to read any creative ideas in the comments.
We’ll be back next Monday with the solution. Do you know a cool puzzle that you think should be featured here? Message me on Twitter @JackPMurtagh or email me at [email protected]
Solution to Puzzle #22: Cheryl’s birthday
Were you able to solve last week’s puzzle from a Singaporean math exam? I’ve repeated it below:
Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
Cheryl then tells Albert the month of her birthday and Bernard the day of her birthday separately. Then the following conversation takes place:
Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard also does not know.
Bernard: At first I didn’t know when Cheryl’s birthday is, but I know now.
Albert: Then I also know when Cheryl’s birthday is.
When is Cheryl’s birthday?
Answer: Cheryl’s birthday is July 16th.
Shout-out to the wonderful editor of the Gizmodo Monday Puzzle, Rose Pastore, for solving her first puzzle in the series. [Editor’s note: Yep, I failed to solve all 21 puzzles before this one. Feel free to roast me.]
Albert is told a month between May and August, while Bernard is told a day between 14 and 19. If Bernard had been told “18” or “19,” then he would instantly know Cheryl’s birthday (June 18th or May 19th) because those days don’t repeat on her list of 10 possible dates. Albert knows that Bernard does not know Cheryl’s birthday from the day alone. How could Albert, having only been given the month, be sure that Bernard wasn’t given “18” or “19?” Albert must have been told “July” or “August,” which don’t have the 18th or 19th as options.
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
Next, Bernard announces he knows the birthday. If he had been told “14,” it would still be ambiguous to him at this stage between July 14 and August 14. So Bernard must have been told 15, 16, or 17.
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
Now Albert also knows the birthday. If Albert had been told “August,” he still couldn’t distinguish between August 15th and August 17th, so the answer must be July 16th.