I finally finished this weekend’s Listener puzzle, “Three-Square” by Elap, during my commute to work this morning. It’s a cross-number puzzle this time — four times a year the Listener puzzle is a numerical puzzle of some sort. Some of the clues in this one involve finding Pythagorean triples and Heronian triangles — the former are sets of three whole numbers such that x² + y² = z² (thus being the sides of a right triangle whose sides are all whole numbers, like 3-4-5) and the latter are triangles, not necessary right triangles, whose sides are all whole numbers and whose area is a whole number.

It took me a while just to find a place to break into the puzzle, and then once I did I put off dealing with the clues involving Heronian triangles as long as I could. Everything else I could work on easily enough with just a calculator, but the Heronian triangles would be too tedious to attack without heavier guns. Still, once I’d solved all the other clues but those, there was nothing else to do but tackle them. So I opened up a spreadsheet program (I use Numbers on the Mac) and set it up so that if I put the sides of a square into the first three cells of a row, the fourth would give me half the perimeter (the sum of the three sides divided by two), and the fifth would give me the area (if s is half the perimeter, and a, b, and c are the three sides, then the area = s × (s−a) × (s−b) × (s−c)). That made it easy to try out lots of different possibilities without having to calculate that terrible formula over and over and over again.

I finished all that on Saturday evening, but there is one last step. There are still some empty squares in the grid, which we have to complete so that all the rows and columns are “thematically consistent”. Ten rows and columns are completed at this point, each containing either nine or eleven digits, so we have to figure out what these sequences have in common and then complete the grid so that all the rest of the rows and columns have that in common, too. I had fun solving the puzzle up to this last step, but really, trying out various ideas on a bunch of nine-digit and eleven-digit sequences to see if any of them worked out was kind of tedious. We don’t even know whether they’re to be interpreted as single nine- and eleven-digit numbers, or as series of two or more shorter numbers — more Pythagorean triples? more Heronian triangles? — or what.

When I finally found the pattern that all ten rows and columns fit, it looked at first like it was going to be impossibly tedious to do the calculations needed to fill in the rest of the grid, but I worked out a shortcut with the spreadsheet program and it wasn’t too bad. Still, that last step seemed more of a slog than a nice surprise.

Fun but tough puzzle other than that, though.

This morning, a little before noon, I finally finished this weekend’s Listener puzzle, “Liberty Bell”, by Pieman. A fun puzzle but a bit of a workout, as it’s not easy to fill the grid and then there’s still more to figure out after you have.

Part of the puzzle is that only a small number of the bars that separate words are given. We have to fill in the rest of the bars as we solve. This part is not difficult, as the word lengths are given as usual with the clues. But then, once we’ve filled the grid, we need to erase as many of these bars as we can (but not any of the bars that are given at the start) to make new, longer words.

So if you had SPORT in the grid, and the next letter after it were a Y, you’d erase the bar separating SPORT from Y to make it SPORTY. You also do this with whole words: If you had TAPES next to TRY, say, you’d erase the bar between them to make TAPESTRY. We have to do this wherever possible, and it’s possible in quite a lot of places. When we’ve done all that, we get a short quotation out of it, and then we have to make a few more changes to the grid to reveal a “refrain”. Took me a while to figure out that last step, but when I finally did, the result was a nice, silly surprise.

Now that I’ve solved the whole puzzle and know that the final result is, I can see why the constructor had us fill in the grid and then erase some of the bars. There was a puzzle by a well-known constructor from, I think, the ’70s, in which a similar gimmick was used, but it was possible, with a little imagination, to figure out what the final answer was going to be long before you’d filled in the whole grid. With this puzzle, there’s just about no chance of that. I think you really do have to work through the whole thing, solving all the clues and then erasing bars to create longer words and so on, to reach the final answer.

However, this also has the disadvantage of giving us an abundance of three- and four-letter words to actually solve, with the longer words only to be revealed later as the bars go away. More longer words and fewer short words in the first place would have made for more interesting solving. Ah well, I guess you can’t have everything. It’s certainly a remarkable piece of construction.

Two odd little points: There isn’t anywhere on the entry to write down the short quotation; I suppose you could in theory solve the puzzle and submit your entry without ever knowing what the quotation is. (Unlikely, as the quotation isn’t hard to figure out when you’ve removed all the right bars, and you have to remove all the right bars anyway or you’re not going to end up with the right answer at the end.)

And I believe the horizontal bar across the top of the square containing the number 44 shouldn’t have been given to us; it looks to me like it should have been one of the bars we have to fill in for ourselves. It doesn’t affect the result, though.

I finished this week’s Listener puzzle, “News Quiz”, on Saturday evening, but I only just now noticed, as I was preparing my entry to send in, that there is a little extra surprise concealed in the finished grid. The instructions tell us to highlight one “thematically located” cell in each column to produce the answer to a question; they don’t mention that another twelve letters in the grid spell out something more, also appropriate to the theme. Sweet!

A fairly easy and straightforward puzzle as Listener crosswords go, with a lot of clever clues and some nice discoveries to make at the end when the last pieces fall into place.