This week's Sunday Puzzle looks like a mathematical equation. Well, it is a mathematical equation. But hidden in that equation is a poem. Can you recite it back to us?
We're traveling again this week, so we're going with something a little more straightforward than usual. I say "straightforward," but I've seen this equation stump several seasoned puzzle-lovers. Remember, the poem in question is a limerick, so head here if you need a refresher on their meter and rhyme scheme.
We'll be back next week with the solution – and a new puzzle! Got a great brainteaser, original or otherwise, that you'd like to see featured? E-mail me with your recommendations. (Be sure to include "Sunday Puzzle" in the subject line.)
Last week, I asked you to decipher a message hidden in a star:
The numbers from 1 to 16 were written in the circles of the diagram below in such a way that the sum of any four numbers in a straight line was the same. Then the number 1 was replaced by the first letter of a saying, number 2 by the second letter, etc. The final configuration is shown. What was the saying?
At the time, I told you the problem could be approached one of two ways. The first way was anagrammatically. This is how I solved it. I toyed with letter combinations for about 30 minutes before piecing together the saying, which is:
Do a good turn daily.
From there, I backstepped my way through the problem to find the the sum shared by each of the 8 rows (34), and the distribution of numbers in the star. Seven of the letters (G, T, U, N, I, L, and Y) appear only once, and so their numbers can be assigned to the star pretty straightforwardly. The rest can be deduced through summing and Ariadne's thread (it's unclear how commenter Sillvva came up with this solution, but wisethesimple seems to take an approach similar to mine, here):
This, I thought, was the first of two approaches. "The way out," as it were, for those less experienced in the ways of number theory and geometry, both of which Michelle G. – who submitted the puzzle – said were necessary to decipher the star without brute-forcing the anagram angle.
In looking through the comments from last week's puzzle, it seems this may in fact be the best approach to this puzzle, whether you're well-versed in number theory or not. Several of you correctly deduced the shared sum (this puzzle is a variation on magic square puzzles, which are amenable to a similar approach), but quickly encountered a road block: The numerical solutions to the star-grid are numerous. Very numerous. So numerous, in fact, that, in the absence of one or two pre-placed numbers to "seed" one of the rows, the problem appears to become unsolvable – at least by hand. As commenter JonathanPonikvar put it: "It's like handing someone a blank Sudoku board and asking them to find one specific layout."
So is this puzzle unsolvable from the numbers angle? Not exactly. Rather, it seems to demand a programmatic approach. Commenter loveboys summarizes the situation as follows:
I wrote some code to generate all possible combinations.
Altogether there are 1792 combinations, linked below:
The numbers are listed in clockwise order starting from the 'O' at 7:00. Each unique combination will appear at least 8 times (4 x 2 rotated and mirrored).
After attaching letters to numbers, I get the list below:
Then it becomes simply a matter of pruning away letter combinations that don't work, searching for possible words, and eventually I got the solution on line 1206.
Line 1206, of course, reads:
I admit, I found this numerical approach to be inelegant and unsatisfying. I do my best to pick puzzles that can be solved without experience in coding (not that there's anything wrong with programming puzzles – in fact, many of our puzzles lend themselves to such an approach). And while the anagrammatic angle does not require numerical analysis, my phrasing of the puzzle last week made it seem like the two methods of analysis were on more even footing – and for that, I apologize. It is my general policy not to post problems until I, myself, have solved them. And while I had solved this puzzle with one approach, I did not take the time to verify that the numerical alternative could be performed without resorting to code. I took Michelle G.'s claim that the problem could be solved from two angles at face value, without verifying her solutions.
In fact, this week, when I asked Michelle G. to supply a numerical solution, she clarified that she, too, had taken an anagrammatic approach. She, too, assumed that a numerical solution was possible. Baffled, and in search of answers, I tracked down the original solution to this puzzle, which appeared in a 1971 issue of MIT's Puzzle Corner. There, I found this:
"No one gave his reasoning," writes MIT Puzzle Corner editor Allan Gottlieb, "but three people, in addition to the proposer... agree that the saying is 'Do a good turn daily.'"
So here's the question: Is there, in fact, a way to solve this puzzle from a numerical angle that does not require uncanny luck, or a background in programming? I'm posing the following challenge: If such an does approach exists, the first person to email me his or her reasoning will receive a free copy of How to Solve It, the classic text by mathematician G. Pólya considered by many to be the definitive guide to mathematical problem solving.
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