# Three-Square

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 × (sa) × (sb) × (sc)). 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.

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