The 4th Dimension, from the mind of Mash

I hope it is not inappropriate to write up a Listener puzzle in this forum, but it’s the closest thing I have to a ‘home’ in the crossword cloud. Plus most of our regulars will have done it (I hope), so won’t feel left out. And finally, if it entices an outsider or two onto our pages that has to be a good thing.

In my latest editorial I wrote about the elusiveness of the PDM. I can stare at an intractable puzzle with no ideas in my head, then suddenly have the solution, but often only seconds later I am left wondering what exactly got me there. Setting can be similar. In this particular case I am not helped by the time elapsed since compiling, but I am struggling to put my finger on exactly how I arrived at the form it finally took. The initial germ of the idea arrived when testing Twentysix’s Cap and Bottles for Magpie 79, whose Klein I initially assumed was Felix rather than Yves. I then thought back to the great Elgin and his Carte Très Blanche (Listener 3713) and wondered whether it would be possible to take that extra step into the fourth dimension. I didn’t, and still don’t, know of a Klein bottle having been attempted before, so the challenge was there to be grasped. Using an unbounded continuous surface would give scope for so much more symmetry, of which I’m a big fan. Dimitry’s Torus Spiral (Listener 3852) impressed me greatly at the time with its cascading eight-letter entries and I hoped I could do something along those lines.

So to the drawing board. Several problems immediately rear their heads. What form will the entry take? Elgin managed to persuade the editors to accept home-made Möbius strips (for which top marks, innovation gets plenty of credit from me), but a Klein bottle is perhaps a step too far even for Listener solvers. Also, without stepping back into that fourth dimension for a clear view, what is going to distinguish my grid from, say, the torus, which has appeared with great frequency over the years? To a two-dimensional surface-dweller who moves in any direction and finds the landscape under him repeating after a given interval whichever way he goes, there is no distinction between many possible ‘universes’. It could simply be infinite and repeating, or curved back to meet itself in a number of possible ways. What our flat man needs is a way of looking ‘up’, and/or measuring the curvature of his surroundings. But all far too theoretical for a crossword.

Elgin presumably had a similar problem in that his long thin grid could just as well have formed the outside of a band. He got around this cleverly by having his down clues wrap over the edge, thus fixing the position and orientation of the ‘back’ at any point. Like a Möbius strip, a Klein bottle is just a surface with no ‘inside’ and ‘outside’, so can’t be filled like a sphere or torus. But unfortunately it has no edge, so the same trick is not available. Hmmm. But it does mean that the ‘back’ is always used. So what if our flat friend could look ‘down’ and see what was under his feet? That would alert him to the possible closed nature of his universe. Then by leaving markers and walking half a ‘repeat’ in the right direction he’d be able to see them on the ‘other side’ and deduce its exact form.

So far so good, but how do I split it to reduce from four to three or preferably two dimensions? The geometry of non-orientable surfaces was never entirely at the forefront of my knowledge, even in my mathematical days, so a quick Google for some useful properties comes up with the fundamental polygon. Obvious, really, as that is what all setters are (probably unwittingly) using when they wrap entries around to the other side of their grids, but giving it a name in my head helped a lot. This construct is not only a great way of identifying topological differences between closed geometric surfaces, but is flat and square – like a crossword – perfect.

Next problem: do I insist on a double-sided entry? For the Magpie, the intended audience at this stage of genesis, it’s theoretically no problem. For print subscribers we send out entry forms with blank reverse sides (technically known as ‘cleanbacks’ in households with small children who reuse junk mail for scribble pads). Electronic subscribers print their own, so that’s fine. But then quite a few submit via email, either scanning or creating on screen from scratch, so where would that leave the two-sided grid? Hmmm again. Now comes a crazy thought the like of which pops into my head from time to time but rarely bears fruit – what if the ‘bottle’ were actually made of ‘glass’? Then half the entries could be written on the ‘back’ in mirror-writing, thus giving a nod to another Elgin favourite of mine – Asylag.

Nice idea, which of course throws up further difficulty. I’ve now created another grid as part of the solving process. There was already the infinite repeating pattern, then the smallest non-repeating ’tile’ of that to be split on two faces, now there’s the transparent conjunction of the two. The page is getting full and I’ve not yet placed a single entry. Setting-wise there are issues too. It is of course going to mean leaving blank cells in the initial grid or the final one will look a fright. Or could I somehow use symmetrical letters only? No, too restricted, and anyway then the point of transparency is lost. Or overlay letters to produce different ones? Very unlikely. A pair of Cs (or with vertical mirroring two Us) makes an O. Two Js is U. F and L (vertically) make E, two Ps a B, M becomes W, but very little else is useful. Maybe just a few could back on to each other. If I could somehow get a third of letters to work back-to-back, then by leaving a third of the repeating grid blank the whole thing would be filled by the end. But how and why would cells be left blank in a barred grid? Unsignalled in wordplay perhaps. Or just too short to fit entry spaces? The checking would have to be spot on to make that work, so that answers wouldn’t ‘slide around’ in their spaces. It’s all getting a bit convoluted, and there’s still no thematic justification. But wait. Isn’t there a ready-made grid type that leaves blanks as standard? What percentage of cells in a blocked grid are blacks? As a minimum there are the 49 (alternate rows and columns) plus say one for each row and column with entries, making 65 of 225, or 29%. Most grids would have a few more where odd cells are left over when entries don’t meet properly, plus of course in an infinite grid there would be an equal number of entry and non-entry rows and columns, adding a few more towards that guess of a third.

But, and it is surely a massive BUT, can there possibly be a grid that reflects onto its own back with no gaps where blocks meet each other, and preferably no places where checked cells meet each other (finding a fill with symmetrical letters checked four times would surely be going some)? Could it be done at all, never mind with symmetry, the absence of which is really not an option to me? The only way to find out is with experimentation. Why not think big to start? I’d ideally like rotational symmetry, and if the repeating unit is a square there could easily be some reflection in there too. Now, the way the fundamental polygon works is that in one direction it joins in well-behaved fashion to form a cylinder, but the other way incorporates a half-twist to make a Möbius strip. Thinking about either of these in isolation is easy, but both together turns up the pressure on the brain cells. So for crossword purposes we’ll think separately. [For those interested, in my head I do the cylinder first then stretch it round like a hose. but rather than make the simple torus join, make a quick trip to a convenient higher dimension in order to get 'inside' the other end of the hose, and finally turn one end inside-out to make the join. There are other clever constructions, like sewing a disc onto a Möbius strip, but I can already feel my brain overheating.]

Now, the ‘standard’ size for a barred grid is either 12×12 or 13×13. It is amazing how obviously different anything else looks once one has done more than a few. I don’t really want to be a long way off that for solver comfort. Clearly a blocked grid can only repeat after an even number of cells, so my polygon has got to be even in the ‘cylinder’ direction. Also, for blocks not to back against each other, it’s going to have to be odd in the ‘strip’ direction. Therefore not square. But there is some immediate good news – this configuration actually forces the alternate-row blocks (which have no freedom to move) to back neatly against all the checked cells, making life easier. Still thinking idealistically though, I’d like a square repeat. So my polygon will be some odd multiple of half its ‘cylinder’ dimension in the ‘strip’ direction. The only way to have a square repeat when filled is for that to be precisely half, but to get a reasonable size it looks unlikely. What about one and a half? My filled repeat is then three grid repeats next to each other. With a 10×10 pattern repeat and a 10×15 final grid, that would fall nicely in the desired range. Coming along smoothly, but 10×10 is still very little to play with in a blocked grid format if everything is going to repeat precisely. In fact, nine-letter words with a single block in every (alternate) row and column is the only really desirable option, as all others either result in words too short (two fours with two blocks) or in over-blocking and/or over unching (seven, say, with three blocks). I have enough experience of setting to know that I always over-restrict myself and have to make compromises somewhere. Already this is looking strongly like one of those cases. But after a while a possible pattern emerges with no blocks backing onto themselves. Wow and double wow – it is possible. But some entries, taking into account ‘reverse checking’ are completely checked, and others have two unches. In theory, given the number of blocks, it should be possible to have each entry with one unch only. Plough on, more in hope than expectation, until finally a beautiful configuration pops up like a mirage from the desert. And on top of that it has two-way mirror symmetry and four distinct centres of 180° rotation symmetry – joy. [Needless to say, the preceding sentences took many frustrating hours of experimentation before coming together more neatly than I could possibly have hoped. Looking back now, each new revelation about the symmetry and possibilities gave me more insight until finally I should have been able to devise a logical system for exploring all eventualities. But hey, I had a pattern that worked and grid filling beckoned.]

Next stage, populate those cells. For a naïve nanosecond I think it might be possible to do so with real words. Who am I kidding? Nine-letter words with a single unch each and nearly half of the letters restricted to either of the sets (AHIMOTUVWXY) or (BCDEHIKOX). When I say ‘half’, the final figures are 7 blocks, 7 unrestricted (backing onto blocks) and 6 symmetric for every 20 cells of blocked grid. That’s very close to my original thirds conjecture, so I’m pretty chuffed. But it means something has to give in the fill, and I plump reluctantly for jumbles, most likely all acrosses or all downs. And of the two sets of letters the first looks the more friendly, which decides me in favour of a vertical grid format (cylinder when vertical edges join, Möbius for the horizontal). At this point I come back to an earlier thought – that I want something to be found in the grid to describe the theme. It is unlikely but conceivable that a solver might realise that half the grid needs to be stuck on the back without actually getting the point. As a solver I’m not a huge fan of puzzles I can complete without full understanding, but as a setter more so, and generally I’m mean enough to ensure that failure looms for the non-conversant. The most obvious is KLEIN BOTTLE itself. Just too long to run across in one string, it could run down a column, especially if entries are jumbled. But then it would be visible before ‘sandwiching’ which I’d like to avoid, and might put too many restrictions on one or two entries. What about diagonal? If it crossed black cells in the original grid it would take alternate letters from front and back, thus be very hard to spot before the crucial step. Even better if the O sits in a cell where front and back coincide it could be made of two Cs. Perhaps even if I choose the trickier of the two letter sets I could do the same with E and B as well. But sadly it’s impossible running diagonally over the black cells as nothing will be a front/back mixture. After stacks more time experimenting I reluctantly concede that something must be canned and go diagonal, spending yet longer trying to position reversible letters on the back face so that it reads correctly throughout. In the end the best I can do is all but L and the final E right-reading.

Plenty of other nice snippets could go in. Some background reading comes up with the lovely detail that “bottle” might be a typographical error. The German word for ‘surface’ is Fläche. Write that on a typewriter with no umlauts and you have to put Flaeche, which could easily be misspelled as Flasche, the infamous bottle. So obviously I spend some time pondering how I could use that somehow. In the end it finishes up with the “Klein Bagel” and various others on the scrapheap of ideas that mounts during the setting of any puzzle.

So finally I stick my K top left, put N and B above each other in column 5 and run diagonally down to hit the right edge just above the bottom. It can’t nestle in the bottom corner because of needing to cross blocks, so it’s about the neatest I can manage. It would have been nice to use that bottom row so that the highlighting could ‘fix’ the positioning of the submission grid. The problem, in the back of my mind all along, is that theoretically the bottle could be cut at any point, allowing any cell of the original 300 to be top left, before even considering rotations. If I give the clues in ‘normal’ grid order, then the blocked grid is fixed to within 2 rows and 3 columns, with most solvers likely to put cell 1 top left. Even that is not a given, and anything could happen during the translation, so I’m really going to have to say something like highlighting starts top left and reads left-to-right. Wait a minute, though, two of my letters are backwards, so actually they could be regarded as reading right-to-left. Whilst I may be obstructive to any solver who hasn’t seen the light, I want to make sure that those who are right definitely know it. So I could say “partly right-to-left”, which rather sneakily captures the truth that the words actually run left-to-right (and the submission grid therefore can’t be entered from the back), but there’s something else going on. I decide to dismiss the possibility that anyone would enter it upside down.

I won’t bore you with the trauma of the grid-fill except to say how often the same words pop up when characters are so restricted, and that even with completely jumbled downs my ludicrously high self-inflicted checking makes it near impossible. Eventually I manage an acceptable version containing real words without too many dodgy proper names / foreign words / archaisms etc, and breath a big sigh that it has proved possible at all. I still have a lot of -ATE endings and some pretty obscure words, but I forgive myself.

All along I’ve been pondering various chunks of preamble wording, but now I have to decide exactly what to get the solver to do, and therefore whether I need any clue gimmicks. Also how do I signal the down jumbles? Theoretically of course they don’t need anything, as in the final reckoning they only have one unch each and are therefore forced. But even with my evil setter hat on I can see that it would be pretty tough going to make the first grid under those conditions. Unless perhaps I give the grid itself, but that would take too much space and spoil some of the fun. Could I make the wordplay of the clues lead to the entry? After some thought two objections come up: first the strings are so horrible that I’d be forced to use constant acronyms (anagrams obviously would be out), and second it would make clues much harder as partial wordplay would not lead to definition-guessing as often happens with my own solving. What about just indicating the positions of the five checked letters in the blocked grid? Possible, but would lead to much more solver agony over how to laminate the two halves and reduce the effectiveness of the pdm moment for the successful solver. A first draft gives alphanumeric differences between successive letters, but I feel that it’s just obfuscation for no reason, so in the end I plump for providing the order straight as a digit string. Pretty ugly, but I hope it will be forgiven once the solver sees the light. So it looks like my clues are to be normal. I always prefer that, but do I need anything else to get the message across?

I decide that a neat finish to the puzzle will be to ask the solver to complete the grid using two pairs of coloured arrows. On its own a tantalisingly ambiguous requirement, but after grasping the concept not too onerous, surely? For that they’ll need to fully understand the fundamental polygon concept so those words will need to be hidden somewhere. I’ve got 30 clues, so that leaves 12 letters spare (I don’t hold with using only some clues unless there’s good reason). ‘Fourth dimension’ would be nice, but it runs over, so maybe abbreviate 4th. But then I can’t use initial letters of extra words, and first letters of clues would look a bit odd. I can’t now remember where it came from, or even in which order, but around this point the various desires to hide a message, to ‘fix’ the grid position, and not to miss out on the opportunity to combine reversed letters all came together in another insight. Two back-to-back Ss make a perfect 8. Put that in column 8 and the ‘horizontal hold’ issue is sorted, plus I could hide a message in the 8th character of clues which is far enough in not to be immediately visible to a casual acrostic scanner. And if I call the the S-S 8 a ‘clash’ it will remain an enigma to those still searching, but further confirm successful solvers in their endeavours. It means not a little grid rejigging but is worth it for the payoff. It’s starting to look good.

Clue writing is the usual mix of a paltry few beauties that suggest themselves immediately and the rest prised out after much work. I can’t resist 15d once it comes into my head, and of course the intended audience is still at this stage the Magpie subscriber. The apostrophe that shifts the M into eighth spot just nudges me towards being sneaky and including punctuation (maybe spaces is going too far). Better still, the message is now likely to remain hidden until specifically sought. The constraint does hinder somewhat, but certainly focuses the mind. It is the best of both worlds in a way. As far as the solver is concerned the clues are normal, but the extra work by the setter provides a bonus discovery later. I have a tendency to overuse the punning definition but 8a really appealed to me, and I hope the definition in 10a makes up for the convoluted wordplay.

A preamble is then written which in most respects resembles the final version except that the “… thinking it over…” and “…correctly seen through …” stuff starts out as: “Blocks must not be blacked in for reasons to do with the grid which should be clear.” Also “…reading partly from right to left” starts out as “…second and final parts read backwards.” Crucially and fatally though, no clue numbering is given, although notice is served that they’re in conventional order. On top of that there is no indication of the size of the blocked grid repeat.

Finally I’m ready for a tester. It’s been over a year since our launch editorial issue in which I had two puzzles, both tested by my usual source. But for some reason I see this as my first ‘proper’ Magpie submission since the new editorial team devised the submissions process. So I give it to the official first vetter, the newly-crowned UK Crossword Champion (that’s how long ago it was) Mark Goodliffe. I judge it hard, but surely if anyone can do it … But sadly, it seems I’ve gone too far. A committee of cryptologists might eventually have got there, but not a single solver. In heroic fashion Mark solves 27 of the 30 clues cold, and those in the ‘raw’ state before his improving tweaks. How he does these things will remain beyond me, and his competitors in the Times Championships too, it seems. But with no indication of any spatial order he is stumped on the grid pattern. Mea culpa – I’ve spoiled it for the first solver. I give him hints, enabling him to complete grid 1, after which he finishes and generously enthuses about the concept. We decide to include clue numbers which goes a long way towards providing structure. I also (reluctantly) agree to divulge the dimensions of the pattern repeat. My protestations that length plus quantity of answers should lead solvers clearly to a 10×30 or 30×10 formation fall on obstinately but correctly deaf ears. I’ve spent so long breathing the symmetry that it’s obvious to me, but Mark argues that too many people would have their enjoyment spoiled for which a few delighted achievers would not compensate. The final arrows requirement is another casualty, with the justification that it’s too maths-theoretical for most people and anyway requires Wikipedia. I’m sad, as it means that solvers can finish without finding my painstakingly concealed message, but it’s not the end of the world.

So to the other editors. Positive responses, minor tweaks, until Simon throws a spanner in the works by insisting that the puzzle should receive the wider audience that the Listener affords. Chris argues strongly against on the basis that the Magpie should not give up its best puzzles like a poor relation. I abstain in case hubris interferes with impartiality, and the others side with Simon. Remember this is in the days before Shane could have been accused of any conflict of interest. Those in the know are aware he still can’t, of course.

Extremely flattered, I send off my submission and wait to hear. It means coming up with something else for the ‘barless’ issue 79, but that’s not part of this story. Time passes. I hear rumours. Finally it comes out that in a period of editorial transition a black hole has opened up, sucking large numbers of aspirant puzzles into its maw, apparently even those which have recourse to higher dimensions. Roger takes over as first vetter, but apparently it’s still politically delicate to ask after the Bermuda Triangle. Sadly Derek Arthur then dies suddenly, Roger moves up, appoints Shane and the whole team has changed within a year. The trove is finally unearthed, and Shane is given his quota towards the great catch-up. Spotting one he can get off his desk relatively swiftly as he’s already tested it, he passes it on to Roger. Roger then does the same as Shane did originally, which is to reverse the order of columns individually instead of flipping the whole bottom half. It means no reversed letters and no clash. I confess it’s not something I thought of in compiling, but it is very hard for a setter with the answer already in their head to anticipate everything solvers will do. Editors are much more likely to spot such things, and it is agreed that more hints are required in the preamble if it is to be a goer. Several suggestions ping back and forth and we thrash out a final wording. I make concessions, some reluctantly, but the place I make a real stand is when the proposal comes to say ” … the second and final LETTERS read backwards”. To my mind the main pdm of the puzzle is the reversed writing generated by using the ‘back’ of the polygon, so that mustn’t be mentioned explicitly. I win that one. Given the preamble changes and editor failure on draft one, a further tester is employed, so at my end things go quiet. But then great news. It has been accepted and awaits a scheduling spot far enough removed from the mini-glut of cartes blanches over the summer.

I suspect that falling between the cracks for nearly two years (not as long as some, I hear) actually benefitted me in that a first vetter coming to it completely cold might have rejected it out of hand as too obscure. Who can tell. There are and always will be conflicting opinions on whether the Listener is too easy and still dumbing down, or too abstruse and removed from the grasp of all but the elite. The only undisputed fact is that the number of all-corrects these days is vastly higher than it was a generation ago, but I suspect that has far more to do with online help than puzzle difficulty. Everybody knows what sorts of puzzle they prefer, and as long as they are open-minded enough to try something different when that’s not available then the series will flourish. Speaking personally, there’s no point in my trying to set a puzzle that I wouldn’t get pleasure from as a solver, so Mash is never going to be a pushover. If I sail too close to the Listener wind, hopefully the Magpie will pick up the wreckage. Since this puzzle was conceived I have at least become more prolific, with more than two puzzles a year in the Magpie. Sadly I’ve had no time in the past few months, but I certainly intend to continue in the same vein as soon as the opportunity affords. I hope there will be some more Listener offerings as tasters, but if certain voices in the boardroom gain the ascendancy then the best will be kept for the Magpie. What better reason could there be to subscribe?

Finally I must offer heartfelt thanks to all those who so generously praised the puzzle, in the postal feedback I received from John, via email and on the various messageboards. I’m still inexperienced enough that getting a puzzle to work at all is a real kick, having it published doubly so, and getting positive feedback triples the whammy. Thank you everyone.

2 Responses to “The 4th Dimension, from the mind of Mash”

  1. Richard Wells Says:

    It did occur to me, when solving last month’s two torus-shaped puzzles, that, for variety, one of them might have been designed on a Klein bottle. I didn’t send that comment in, because I thought a puzzle on a Klein bottle would have been too hard to devise.

    But Mash, for whom it seems nothing is too hard, had already made a Klein bottle puzzle! I am really sorry that it did not appear in The Magpie, where I would have seen and tried it, but only in a lesser publication which I don’t subscribe to. Is it now posted somewhere on the internet, where I can have a look at it? Mash’s piece here gives I think neither title, date, number nor other reference.

  2. David Agg Says:

    I meant to say this earlier, but this seems as good a place as any. I didn’t find Mash’s hidden message, not least because I thought the message was COLUMNATE (ie COLUMN 8), which seemed a good description of what you had to do to make the grid into a Klein bottle.

    I’m quite happy that the correct answer is better, but every so often I find what seems a satisfactory solution and then stop looking further. The worst recent example was in Elgin’s Listener “Change of Clothing”, where I thought that TROUTERS was a satisfactory answer to “wrong trousers”, and so failed to find (or even look for) Wallace and Gromit. Sometimes local optima give very good impersonations of global optima.

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