Kominers’s Conundrums: Back to a Puzzling Future

(Bloomberg Opinion) -- Just over a month ago, I revealed that our Conundrum was completely hidden, then urged solvers to chase clues inside the column’s text. But I didn’t reveal everything. Indeed, I’ve been secretly hiding more clues for a month – one in each of the past four Conundrums..So let’s take a trip through time and revisit our code-breaking, coin-weighing, dinner party-saving, and chocolate bar-splitting Conundrums. If you find anything that looks unusual, it could be a clue..And here’s one more (slightly mysterious) clue that should help you figure out how to put the other four together into two words, which are the answer:.Alright, ready to start solving? Go for it! Let’s do the hidden puzzle again! .If you manage to unwind the answer – or even make partial progress – please let me know at skpuzzles@bloomberg.net before midnight Eastern time on Wednesday, June 17. (If you get stuck, there’ll be a hint announced in Bloomberg Opinion Today on Tuesday, June 16. Sign up here.).Note to analog solvers and/or those reading in syndication: While solving this puzzle does not require a Bloomberg subscription, it does require looking at the versions of each column posted at bloomberg.com/opinion..Last Week’s Conundrum .We took a crack at breaking a world record – breaking up a world-record chocolate bar, that is. With 38,324 squares to separate, at first the puzzle looked so large as to be beyond comprehension: Should you begin by splitting the bar in the middle? Or towards one side? And is it better to break lengthwise or widthwise?.Many solvers tried experimenting with smaller bars, and quickly noticed a surprising pattern: The number of breaks needed doesn’t seem to depend on the sequence you choose at all..For example, suppose we start with a 2 x 3 chocolate bar. If we first split widthwise, to get two 1 x 3 pieces, then we have to break each of those twice to fully separate all the squares – leading to five breaks in total. If instead we start by breaking the 2 x 3 bar into a 2 x 1 and a 2 x 2, then we need one more break to finish splitting up the first piece, and three for the second – again giving a total of five..It turns out that this pattern extends to chocolate bars of any size: No matter what sequence of breaks you choose, the total number of breaks needed is always the same. And that’s not all – it’s always equal to the total number of squares minus 1..To understand why, we have to switch our perspective slightly. Instead of thinking about the bar’s dimensions, we should look at the total number of pieces..At the start, the bar is in a single piece. Once we break it somewhere, irrespective of where we break it, we end up with two pieces in total. And when we break one of those pieces next (again, no matter where we break it), we find ourselves with a total of three..More generally, no matter what we do, each break leaves us with one more piece in total than we had before..Now we know that in the end, our bar has to separate into 38,324 pieces. Since we start with one piece, that means we need 38,323 breaks to bring us to the proper total..Anna Collins was the first to solve this week. Others among the 22 solvers included Michael Branicky, Ajay Kumar K S, Andrew McDonald, Michael Mongan, Molly Singh, Franklyn Wang, and the Kanner sisters. .I also asked readers to think about the logistics of actually breaking the bar:.Several solvers observed that at one break per second, you would need almost 11 hours to fully separate the bar; taking breaking time to be proportional to the size of the break, Winston Luo came up with an estimate closer to 21.4 hours. Dianhui Ke proposed an especially speedy strategy in which Usain Bolt zig-zags back and forth across the bar with a pizza cutter for 1.7 hours; and Zoz pointed out that we only need five minutes if we allow cuts rather than insisting on full breaks. Finally, Adam Rosenfield sent detailed engineering specs for how use a circular saw and a telescoping boom lift to break up the bar from its original vertical display configuration without creating structural failures – only 26 hours required. .The Bonus Round.Parrots solving puzzles. Play in this weekend’s Sudoku Grand Prix (hat tip: Scott Wu). Hack arcade jackpots and the Legend of Zelda’s squid-spotting game; then try the solitaire board game that beats you every time. “The Puzzling Search for Perfect Randomness”; making magic squares with the card game SET; plus that perpetual energy paradox puzzle had a solution after all. And inquiring minds want to know: How big is infinity, really?.Special thanks to Daniel Chiu and Eric Mannes, who not only solved but also pointed out an important typo in the Conundrum’s original statement. .For bonus points, Elizabeth Sibert noted that if the chocolate bar is 5cm thick, stacking all 38,324 squares on top of each other would make a pile just shy of the height of Mount Mitchell. If we instead put the squares end-to-end, they would stretch longer than the National Mall. .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..©2020 Bloomberg L.P.

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Kominers’s Conundrums: Back to a Puzzling Future

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