Symmetry

I’ve just written about music and crosswords. One of the common features I identified was an interplay between structure and freedom. I’d already started sketching out a blog on one obvious structural feature – grid symmetry – so now seems a good time to dig deeper.

The first thing one finds is the structure/freedom interplay recurring, not least in the profusion of usable grids. Set yourself some reasonable criteria for numbers o words in a 12×12 grid, and the number of unchecked letters they may contain (e.g. one in a 4- or 5-letter word, two in a 6- or 7-letter word, and so on) and how many grids are there? A few years ago someone attempted this calculation for Azed – we’re not talking googols here, but we are in the 10**20+ range. The 15×15 blocked puzzle is probably in the same – rather extensive – ballpark.

So structure again, but so many possible structures that a sort of freedom results. The question of ‘Why symmetry?’ still arises. It’s perfectly possible – though I’d say harder – to meet your length and checking criteria in an asymmetric grid, so why choose symmetry?

Part of it is clearly aesthetic, and rightly so. There is something more attractive about a symmetric grid. However, small lapses in symmetry often go unnoticed (more on that later), and large ones are so blatant that no-one cares. The ones that stick out the most are the ones where a ghost of symmetry remains. These bespeak attempts that narrowly failed, but what they often convey is a sense that the setter simply gave up. My eye is often drawn to tweaking these – ‘you could have had X there’, ‘one thematic entry less and it would have worked’ – and so on. Better to abandon symmetry altogether than adopt such a halfway house.

But tiny lapses seem to slip by. There a puzzle on this site where I had a long thematic down entry, and one example of the theme on each row. The thematic down entry was a 14-letter noun phrase. It could have been pluralised, though it didn’t look quite right and, in any case, I wanted ‘5 + wordplay’ rather than ‘One of 5s + wordplay’. So I shifted one block: the top row was an 8-letter and a 6-letter word, the bottom row two 7-letter words, with the 14-letter thematic running down the central column.

“What do you think?” I asked the editor. “What do you think?” he asked a message board. There was some discussion – nothing especially heated. We waited a few weeks, and slipped the puzzle in.

And scarcely anyone noticed. I suspect the fact that it looked like a normal grid overrode spotting the slight asymmetry. This finding is replicable – I pulled the same trick a few months later, for a perimeter quotation that, annoyingly, had an odd number of letters. I got the same result.

I’ve also toyed with symmetry across the diagonal – now that does get a few cries of ‘Asymmetric!’ followed by ‘Oh, hang on…’ I think it would be hard to get a truly asymmetric grid into an outlet for blocked daily puzzles. They’re commoner in barred thematics. On the whole, I’d always prefer symmetry, and, as a novice setter, I found symmetry a useful discipline in forcing me to consider checking and fairness to the solver.

With all that in kind, here’s a grid. It’s a slightly modified version of a Phi Listener puzzle from a few years ago. After I’d completed it I noticed a curious feature, and the tweaking I’ve done merely took out a thematic entry and adjusted a few bars to complete the effect.

 

It’s clearly asymmetric, isn’t it? Yet every bar has a symmetrically placed partner. How is this achieved? Most of the vertical bars have rotational symmetry (and even the four that don’t could be removed, although that would leave some large expanses of white), while horizontal bars have mirror symmetry. I don’t know how I stumbled on this – it wasn’t consciously designed in – and I suspect it wouldn’t be feasible in a more heavily checked grid.

But is it symmetric or not?

 

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