Finally finished this week’s Listener puzzle last night. It’s called “Square-bashing”. Four times a year, the Listener puzzle is a crossnumber instead of a crossword and this is one of those times. All the entries in the grid are perfect squares, but the clues lead not to the square numbers themselves but to their roots. 20 letters are assigned values from 1 to 20, and the clues are algebraic expressions like TA + INT and ETER + NAL.

To complicate things further, the grid is divided into two halves, left and right. The two halves have the same pattern of bars, and each half is numbered separately, so that there are two of each clue number — two 11 acrosses, two 6 downs, and so on. The two clues at each clue number are given together, and you have to work out for yourself which clue goes with which side of the puzzle. You have to solve the two halves as separate puzzles, and then at the end there’s a way to figure out which half goes on the left and which on the right. Finally, the number that goes along the bottom of each half is left unclued, and you must find two words that can serve as clues for them, using the discovered values of the 20 letters.

It didn’t take me long to see a likely way to break into the puzzle. It had to do with the lengths of the squares; if a clue leads to an answer with four spaces in the grid, for example, then the entry must be a perfect square between 1000 and 9999, which means that the expression in the clue (which is of the entry’s square root) must be between 32 and 99. Particularly helpful is the fact that if an answer has just two spaces in the grid, then it can only be 16, 25, 36, 49, 64, or 81, and its clue must be an expression equalling 4, 5, 6, 7, 8, or 9; not too many possibilities there.

Believe me, I’m not giving much away there. Even seeing this pretty quickly, it took me a long time to actually break in. I think it was an hour of fiddling around before I figured out which letter represented 1. Another half hour or so and I found the letter that represented 6. Only 18 more to go! At one point I hit an impossibility and couldn’t figure out why, and I had to go back and recheck what I’d done. I decided to print out a new clean copy of the puzzle and go over my reasoning right from the start, which of course made it inevitable that I would discover that my error was a silly one in the very last step I’d made, writing down one incorrect digit in a calculation.

I worked on the puzzle on and off through the weekend. Sometimes it went pretty quickly, a couple times it stalled while I searched for a way to break through to the next deduction. On Monday over lunch I finally filled in the last of the entries in the grid, figured out which half of the puzzle was which, and looked for words that could clue the bottom two entries.


For one of the two entries, there was an obvious answer. For the other, there was only one possible way to factor the root into numbers of 20 or less, and there was no way to make a word out of these letters, even with the fact that I could add in the letter that represented 1 any number of times.

I didn’t take the puzzle to work with me on Tuesday, figuring I needed to give it a rest and come back to it with a clearer head. So of course in the middle of the workday it occurred to me what I might have done wrong, but having left the puzzle at home I couldn’t test it. When I got home that evening, I headed straight for the puzzle and redid a calculation. Things worked out as I had suspected they would, and a few moments later I had what was very obviously the intended answer. Very neat.

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