# “Digimix”

Whew. I just now finished this week’s Listener puzzle, called “Digimix” by Oyler. A number puzzle this week, and a very tough one. In order to fill the grid, you have to deduce eleven pairs of numbers, one of four digits and one of five, which have the properties that (a) they contain between them the nine digits from 1 to 9, once each, and (b) the sum of their squares is a nine-digit number that likewise contains the nine digits from 1 to 9, once each. This nine-digit number is broken into three three-digit numbers, so that each set contains five numbers (a four-digit, a five-digit, and three three-digit). The sets then contain 55 numbers in all, and most of these are clued in terms of grid entries. For example, one of the three-digit numbers is clued as b + e + p, and b, e, and p are entries in the grid (which is like a crossword grid but lettered rather than numbered).

It took me about five minutes on Friday afternoon to break into the grid, as the constructor kindly put an obvious clue in the first set of numbers — one of the three-digit numbers is a multiple of another, so between them they must contain six different digits, none of them zero. Other factors limited the values of the two numbers, and it wasn’t hard to run through the possibilities and find the only one that worked. That quickly gave me six squares filled in the grid.

I am giving next to nothing away by saying that much. It took only five minutes to get those, and then by Friday evening, despite a few pages filled with scribbled calculations that didn’t lead very far (for example, I had deduced that the last digit of a certain grid entry could only be 2, 5, or 8, but I could see any way to use this information to get any further), I still had only those six squares filled. By Saturday afternoon, I had a grand total of eleven. (The grid is nine squares by seven squares, or sixty-three in all.) But on Sunday I spotted a couple of deductions I could make that I had overlooked before, and those put useful limits on what some of the numbers in the sets could be.

I ended up getting out the laptop and doing some of the calculations in a spreadsheet program, and that made things much easier; this would be a much more tedious puzzle to solve with just a calculator. Several times I found that I’d narrowed the possible combinations of a few numbers down to six, or twenty-four, or in one case sixty possibilities, and what I needed to do was run the same series of calculations on all of them and see which ones led to possible answers. Much easier to do this in a spreadsheet, where I could set up the series of calculations just once and then quickly apply it to all the possibilities.

It feels a bit inelegant to me somehow that the puzzle needed this kind of brute-force approach (unless I’m just missing the way to solve it without it); but then, at the same time I can’t actually see a rational reason why a spreadsheet program should seem too much to me while a calculator and a printed table of square numbers seem perfectly acceptable tools to me (which they do); they’re all just timesavers for calculations I’m perfectly capable of making with pen and paper alone but don’t want to because it would be long and dull.

But the puzzle certainly wasn’t all about brute force with a spreadsheet. There were a lot of surprising and tricky deductions I had to make along the way, and it was very satisfying to figure those out.

(For those who find this entry because they’re actually trying to solve the puzzle: Yeah, I found the preamble confusing, too, and I think it’s poorly worded. Try changing the first three words to In each of the eleven horizontal rows of clues below. I was confused by the boldface letters at first, too; they aren’t clues themselves, they’re headings for the vertical columns of clues below them. Capital letters in the clues stand for across entries in the grid and lowercase letters stand for down entries. In the final sentence, I am assuming that missing entries should be missing clues because nothing else seems to make sense; all the entries are missing, obviously, because the grid is blank when you start.)