Listen With Others

Keep Your Distance is the first Listener I’ve set and also the first crossnumber I’ve written.

I’m a fairly recent convert to The Listener. Anyone who’s read the Listener guidance on setting numericals may recall the point made there regarding how numericals can attract new solvers and “perhaps some of these newcomers will persevere with harder puzzles or even attempt word-based ones”. This is exactly what happened with me. My first Listener as a solver was L4582 (Full Steam Ahead by Hedgehog in November 2019), after my friend and Listener enthusiast Andy Nelson recommended it. I’d solved a few crossnumbers before but never really a themed one, so the cleverness of construction there was eye-opening. It was also curious to read Hedgehog’s setter’s blog where he thought there were inelegant aspects to the puzzle – I thought it was wall-to-wall fantastic. Since then I’ve become suitably hooked on the “wordy” Listeners too. I’m not the speediest solver yet, but I’m accelerating at a rate roughly proportional to the number of fish with short names in my lexicon.

With an appetite for more crossnumbers after Full Steam Ahead, I subscribed to the brilliant Crossnumbers Quarterly. The issue published near the start of the COVID-19 lockdown suggested readers should have a go at setting if they hadn’t before, so I thought I’d give it a shot. Liking puzzles with visual elements to the endgame and fairly modern themes (although Space Invaders predates me by about a decade), the idea of one of the Space Invaders aliens seemed like it might work. After all, it seems sufficiently iconic of its era and Space Invaders has even found its way into Chambers. Was it too big for a mathematical puzzle? This was a real concern after the CQ foreward mentioned that the largest crossnumber they’d ever published was 11×11 and the Space Invaders glyph was 11×8. Out came a spreadsheet, an 11×8 shaded alien was drawn and the fun began.

It seemed the easiest thing to do would be to put odd (or even) digits in the highlighted alien, with even (or odd) digits in the non-highlighted cells. This would hopefully give a fairly high number of degrees of freedom with which I could try to encode a hidden message. In this message there had to be an instruction that conveyed the highlighting and I also wanted to include something searchable that would help anyone who managed to highlight the little fella but didn’t recognise it. The game’s creator, Nishikado, seemed an ideal word for the latter. I’d seen a puzzle before in CQ that had an encoded message using pairs of entered digits modulo some integer, but rather than saying what that integer was upfront, I thought it would be fun to make the discovery of that integer part of the puzzle (clue “*”). This would also make it harder for people to hack the puzzle by guessing the message halfway through.

I plumped for a four word message with each pair of rows hiding a word. NISHIKADO had to be in the message. So did ODD or EVEN, HIGHLIGHT or HIGHLIGHTED, and DIGITS or INTEGERS or similar. Obviously I had a large amount of freedom regarding exactly where in a pair of rows to place a short word like ODD (3), but a word like HIGHLIGHTED (11) took a full pair of rows. One nice thing about all of those words was that nothing is too late in the alphabet. If we plump for ODD over EVEN, nothing is beyond T, and so we could choose * to be as small as 21. Even if we choose EVEN instead, * can be as small as 23 without destroying the message. This was my first attempt, setting * to be 23, which was in hindsight foolish. Why so? Well, the even/odd nature of each cell in the grid is determined by our lovely alien, so suppose we need to encode the letter H across a pair of cells with the pattern {even odd}. H is the eighth letter of the alphabet, so can be represented as any of the following: 08, 31, 54, 77. None of those follow the pattern {even odd}. This isn’t a problem with some letters (e.g. A can be written 01 or 47 if we need {even odd}, 24 for {even even}, 70 for {odd even} and 93 for {odd odd}).

I quickly realised that * = 25 was the sweet-spot, as any letter (up to X at least) can be expressed as two digits in four ways, one each of: {odd even}, {even odd}, {odd odd} and {even even}. No matter where I put letters in the grid, with * = 25, I would never have any problem like the one I described with H above. I chose to make the alien cells odd rather than even at this stage, just because most of the cells near the middle are covered by the alien and I wanted as few zeroes near the middle to reduce restrictions on the barring. Having decided on the words to make up the message, the row pair encoding “HIGHLIGHTED” was now fully determined as 33595733375/90733955529; no freedom for me here. The “NISHIKADO” pair of rows had a meagre four cells that I was still free to play around with and the top four rows had a reasonable number of “free” cells unconstrained by the message (although still constrained to be either odd or even from the alien’s shape).

At this point, I tried to bar the grid off. I thought barring off a grid fairly full of numbers would be quick and easy, but it wasn’t really, largely due to my complete lack of previous experience I suppose. The zeroes forced my arm in the placing of some bars. Then I spotted a nice way to start if I barred off a few 2-digit clues near the bottom, so I symmetrically barred those and aimed for a sensible distribution of two and three digit entries. I carried on barring in a process that was symmetrical and had a large random component. Then I counted the entries and I had something like 32 down clues… oh dear. I couldn’t imagine any editor thinking that was sensible and I certainly didn’t think it would be a good idea to go beyond 26. For one thing, the lowercase Latin alphabet isn’t long enough!

After briefly thinking the grid was too big after all, I realised I could just chop the corners off the grid to make it smaller. Thank goodness (or Nishikado I suppose) that the alien’s antennae didn’t stick out further. This reduction in size made things just about feasible. With the smaller grid, I started barring off again, this time less randomly and to a greater extent determined by relationships I was spotting between numbers as I wrote the clues. This was quite a fun stage really, albeit frustrating at times. It was the stage that before trying to make a crossnumber I was most mystified as to how it’s done. Having made this one, I’m still really not sure how it works, or at least I couldn’t offer up a convincing algorithm for how I did it. Maybe others have a better bag of tricks, but for me, it was a case of staring at the numbers I’d already clued and trying to tie a few of them together in uniquely defining ways. You just end up staring at a couple of three digit numbers thinking “what on earth do these have in common that somehow relates to another clue or two without being too obvious?”

I tried to keep some degree of elegance in the clues and I think I more or less succeeded with a few exceptions. I like succinct clues like “prime”, I don’t like clues like “sqrt(7a+224b)-2C^3”. I tried to leave clue * until the end of the solving route, or very near the end, so that only those very computer-minded solvers could hack their way to a fair chunk of the solution by brute-forcing the message. The “padding” letters surrounding the words in the message also helped with this, but in hindsight were really quite inelegant. They did give me freedom for my inexperienced clue writing though, so they made my task as a novice a little easier.

With all that done, I did a test solve. It seemed to work, which was something of a relief. What should I call it? Well, “Keep Your Distance” seemed a nice cryptic way to clue Space Invaders without giving the game away and was rapidly becoming in the early days of COVID-lockdown rather topical with there being little talk of anything other than social distancing. I sent it to Andy Nelson, the friend who’d got me into the Listener in the first place, asking him to test-solve. He did so and knocked a few superfluous clauses off some of the clues, so thanks are due to him for this and other useful discussions. I was unsure whether the puzzle would sit better in the Listener or Crossnumbers Quarterly. It felt quite a Listenery puzzle to me, so despite initially thinking I was writing it to send to CQ, I thought I’d try the Listener for this one and see what happens. I was surprised and delighted when Roger pencilled it in to be the next mathematical Listener, given how seemingly topical the puzzle’s title is at present. My thanks to Roger and Shane for their excellent editing.

Writing my first crossnumber was a really fun experience. It used the same skills as solving, with a bit of creativity and extra problem solving thrown in on top. I heartily recommend anyone who’s not tried it before to give it a go; you might enjoy it too!