(Bloomberg Opinion) -- At least in the northern hemisphere, summer is in full swing, with sun-blistering temperatures getting up to 98 degrees and worse. There is only one sure way to beat the heat: a summer playlist puzzle.
If you want to borrow the one on repeat at the Conundrums Cabana, you’ll have to fill in the clues. This week’s answer may depend on your stance on one of the greatest questions in rock-and-roll history. So listen closely!
Love a Battlefield
Than a Feeling
The 59th Street Bridge Song (Feelin')
Should I Stay Should I Go?
December, 1963 (What a Night)
Let me know if you manage to work it out -- or even make partial progress -- at firstname.lastname@example.org before midnight in Highland Falls, New York, on Wednesday, July 15. (If you get stuck, there’ll be a hint announced in Bloomberg Opinion Today on Tuesday, July 14. Sign up here.)
The Prior Conundrum
Faced with four sequences of numbers, the question was simple: What comes next in each?
1, 3, 6, 10, 15, 21, 28, 36, 45, 55 … 66
The first one is called the “Triangular Number” sequence. Its elements count the number of dots in successively larger equilateral triangles, as pictured below. Equivalently, we can think of the sequence as representing the sums of the first N integers: 1 + 2 = 3; 1 + 2 + 3 = 6; 1 + 2 + 3 + 4 = 10; and so forth. 55 is the 10th element of the sequence, so up next is 1 + … + 11 = 66.(1)
1, 1, 2, 4, 7, 13, 24, 44, 81 … 149
Next, we teased the “Tribonacci” sequence. It’s just like the Fibonacci sequence, except we add every three consecutive elements to get the next one instead of adding every two.(2) So for example 1 + 1 + 2 = 4, and then 1 + 2 + 4 = 7. The next number is obtained by adding 24, 44, and 81 to get 149.
73, 97, 1012, 1111, 1126 … 1225
This one, admittedly, was completely different. The numbers don’t have an obvious relation to each other, and thereqs a sudden leap in size: The first two are less than 100, and the rest are over 1000(!). Indeed, this sequence isn’t mathematical at all. It’s encoding this year’s U.S. federal holidays, starting from the date of the column. “7/3” was when Independence Day was observed in 2020 -- since July 4 was on the weekend. Labor Day (“9/7”) and Columbus Day (“10/12”) follow. What’s the next federal holiday after Thanksgiving (“11/26”)? 1225.(3)
3, 13, 1113, 3113, 132113, 1113122113 ... 311311222113
And what about the last sequence, with its long strings of 1s, 2s, and 3s? It turns out to be a bit of a mixture between math and wordplay. Each element of the sequence describes the previous one when read aloud. So “13” describes “3” as “one three.” Then “1113” describes “13” as “one one [followed by] one three.” Then “1113” is “three one[s], one three,” or “3113.” This so-called “look-and-say sequence” was studied by mathematician John Conway in the 1980s. Its elements increase at an incredible pace. The next one is 311311222113.
Noam Elkies extended his first-solver streak, identifying all four sequences minutes before Anna Collins, Zoz, and Max Sabor. Others among the 36 solvers included Elliot Bennett-Spragg, Filbert Cua, James Goll, Kent Holding, Alex Jarzebowicz, Sunil Krishnan, Ridge Phelan Montes, Alex Ognev, Angela Ognev, Jonathan Schachter, Elizabeth Sibert, Clive Sindelman, Robert Tzucker, Richard Yarrow, and David Zhu.
The Bonus Round
A seemingly impossible chessboard puzzle; a crossword with two sides; the Hamilton rap app (hat tip: Greg Llacer). Make your own Bayeux Tapestry (hat tip: Ellen Kominers); explore a gallery of open mathematics; play Pokémon Jeopardy; or recount a giant ring heist. RIP Ronald Graham, the mathematician who discovered a number so large that it’s literally impossible to think about. And inquiring minds want to know: What's that bird sensation that's sweeping the nation (of Canada)?
(1) A quick way to compute the sum of the first N integers, originally due to Gauss, is to add the sum to itself creatively, as (1 + ... + N) + (N + ... + 1) = (1 + N) + (2 + N - 1) + ... + (N + 1) = N * (N + 1). We can then compute that the original sum is N * (N + 1) / 2. And indeed, 11*12/2 = 66.
(2) We usually think of the sequence as starting with 0s before the 1s, so the second and third elements are constructed as 0 + 0 + 1 = 1 and 0 + 1 + 1 = 2.
(3) I linked to a list of federal holidays as a clue earlier in the column, when referencing the fact that Conundrums would be off for the weekend of July 4. Even so, as a few solvers pointed out, this sequence was harder to identify if you're based in a country that lists dates in day-month order rather than month-day.
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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