It’s about 3 years since I came up with the idea for Fisher Transformation so my recollection of its creation is a little bit sketchy. But here we go.
I’ve always enjoyed “recreational maths”, and in particular remember reading some of Martin Gardner’s books as a teenager. I don’t recall him writing about Heronian triangles specifically, but either way he may be the ultimate root cause of how I ended up incorporating some entertaining (I hope) maths into a cryptic crossword. Whilst my first Listener puzzle Pressure Points was Pascal themed, there were no actual “sums” involved in that one (just a bit of drawing).
Having got the idea of Heronian triangles in mind I spent an awful lot of time (and I mean an awful lot of time) experimenting with grid sizes and triangle sizes, long before constructing any actual word content, let alone clues. I think it eventually dawned on me that HERONIAN TRIANGLE was 16 letters long and that a 12 by 16 grid would neatly contain a diagonal of length 20. This gave the opportunity to position a (7, 15, 20) Heronian triangle as I did, but also importantly for that triangle to be derivable as the difference between two Pythagorean triangles (12, 16, 20) and (9, 12, 15). And, of course, it gave the opportunity to somehow get HERONIAN TRIANGLE ending up in the top row. Heron’s formula (see Wikipedia) can be used to calculate the area of this (7, 15, 20) triangle as the square root of (21 x 14 x 6 x 1), but I quite like “96 – 54 = 42”. I thought “including the diagonal from cell 1” was a reasonably unambiguous direction to use the whole leading diagonal, and “part of the top border” ruled out using the entire top border, but maybe the preamble should have been one notch more unambiguous on reflection? At any rate “whole diagonal” + “part of top border” = the unique solution.
At this point I probably started to think about ultimate question too. I don’t claim to have much knowledge of The Hitchiker’s Guide to the Galaxy but I at least knew that 42 was the answer to the ultimate question of life, the university and everything. That connection seemed too good to ignore. And then a little plan began to form. ULTIMATE QUESTION also had 16 letters, and in fact rather neatly had 2 x 8 letters, just like HERONIAN TRIANGLE. Still no words in the grid or clues at this point, but I felt quite elated at the possibility of going somewhere with this!
I think I’d quite enjoyed a then recent puzzle that had employed a cipher, so I started toying with the idea of transforming ULTIMATE QUESTION into HERONIAN TRIANGLE via some kind of cipher (and by the way the transformation via cipher is what eventually led to the puzzle title – whilst Fisher Transformation is something “properly mathsy”, that was just some mis-direction and the title was nothing cleverer than Fisher = Heron and Transformation = use a cipher). Again I spent a lot of time exploring how a cipher might work. For example a straight transformation between the two x 16 letter phrases couldn’t work because of the number and position of repeated letters. I concluded that the best I would achieve was transforming a jumbled ULTIMATE into HERONIAN (both words having just one double letter, so T would map to N). I don’t know whether this was self-evident to solvers or not, but the jumbled ULTIMATE at 7a decoded to HERONIAN, but equally the jumbled ER(ew)HONIAN at 55a decoded to ULTIMATE (so there was a 2-way correspondence for this subset of the cipher). I think this was the point at which I also realised that the only mechanism that could conceivably work was one where a letter or letters were omitted from some of the grid entries. E.g. I would eventually arrive at STEAM IRON for 7a which needed to lose an O. Whilst MINARETS or RAIMENTS might have worked here all else being equal it soon became a juggling act with crossing entries, symmetry, not too much jumbling, and a cipher that achieved real words. Plus I really wanted to have PYTHAGORAS in the final grid somewhere. I’d actually constrained myself quite a lot, even with the flexibility afforded by jumbled answers and lost letters. I’m guessing I ground to a halt about 50 times, due to not being able to satisfy all the constraints. But with perseverance I ended up with a grid populated with legitimate words pre-deciphering (accepting that the jumbling and omitted letters would initially mean non-words) and post-deciphering, or so I thought!
During the trial and error process of constructing the grid a few more angles came into play. One was that I engineered SQUARE LEGS at 13d; whilst this wasn’t blatantly thematic I felt that it was sufficiently thematic in that it could be taken loosely as alluding to the differencing of two Pythagorean triangles (yes I know I triangle isn’t a square, but the square on the hippopotamus and all that!) – in any case it was a gift too good to decline in conjunction with PINDAR / QINDAR at 20a, and the manoeuvring into position of the jumbled PLANCHET at 56a which would decode to QUESTION! Another angle was that I’d begun to mould (and deliberately engineer where I could) the omitted letters to some extent; I at least had an eye on some of the words that could be spelled with them and I’d later end up fine-tuning some entries to achieve the thematic description that was given in the preamble; I also made sure that at worst any multiple omitted letters were consecutive (I think I struggled for days with TELEG(rap)H / TELEGU). The last angle that I’ll mention, because it will come up again later, is the reluctant but sometimes inevitable acceptance of using abbreviations (or in other situations other non-Chambers-supported words); I imagine most setters will settle for this once they’ve run out of other options, and that’s certainly true in my own case; but needs must. One of the abbreviations I settled for in a way was RCOG, however it was also deliberate in another sense. Maybe RCOG is unfamiliar to most solvers, but it’s all too familiar to me because Mrs Snaky is a midwife. Now, Mrs Snaky is totally crossword-intolerant (literally faints if I so much as read a cryptic clue, just to myself not even out loud, whilst she’s in the vicinity), so she is completely oblivious to the existence of this generous gift to her, but the pleasure of a gift can be in the giving so all is well. I’ve had the pleasure of gifting her the inclusion of her organisation as an entry in Listener Crossword 4903, she’s had the pleasure of not having to know about it. I haven’t tried to check but I’d be surprised if RCOG has ever appeared in a Listener crossword before. I digress somewhat.
Many weeks after first pondering how to get a Heronian Triangle into a Listener crossword I had a fully populated, symmetrical grid with acceptable real words, a couple of unavoidable abbreviations, a pleasing cipher and omitted letters that could be arranged to spell something nearly thematic (I’ll come back to this). So I began the relatively much easier process of devising clues.
I wanted to a) give some kind of hidden message that would only appear gradually as clues were solved, and b) not make the puzzle too straightforward. ALEXANDRIAN MATHS ICON was what I ended up with, which I thought might contain enough rabbit holes to delay that particular PDM (e.g. with ALEX, AND, IAN and maybe TH), and I thought that I could make this work with definition misprints. Once deduced it pointed solvers fairly unambiguously to Hero / Heron of Alexandria I think, and given the requirement to draw a shape, soon onwards I expect to Heronian triangles and Heron’s formula to calculate the area thereof.
I guess it was during the clue construction that I realised that 42a sat neatly on the edge of the triangle that would ultimately be drawn (I don’t number my clues until that stage). So that presented a neat and convenient way of asking for the answer. Of course that cell didn’t need to sit on the edge of the triangle to do it this way – it just amused me that it did. And shading that cell rather than asking for the answer to be written below the grid saved a line of print (and thus pleased the editors).
I’m pretty sure that writing the clues didn’t take half as long as constructing the grid, but it was nevertheless challenging. Some of the definition misprints did on occasion lead to some slightly contrived clues that you probably wouldn’t see absent the misprint gimmick. However I think that’s largely OK in the context so long as there’s still some vaguely plausible surface reading. My tiger / tigress clue was pushing the limit. In any case, if the editors don’t think a clue is good enough they will require changes, so that serves as a bit of a backstop.
It was probably at this point, with the puzzle essentially drafted including clues that I tried to look at it as a prospective solver, and asked myself whether I dared to share it with a friendly test solver or two! Obviously I did, but I had at least some reservations – was the combinations of gimmicks fair, was the cipher uniquely and logically deducible, was the number of ambiguous perimeter letters an issue, did the thematic material and preamble lead unambiguously to the required (7, 15, 20) Heronian triangle. Only one way to find out! I am enormously indebted to my test solvers. I won’t embarrass anyone, but I always get constructive feedback and encouragement which is exactly what is needed. As I’m still a relative novice in terms of setting puzzles, I get quite a bit of (helpful) feedback about specific clues where I’ve made some sort of schoolboy error, but it’s also very helpful to see how a fresh pair of experienced eyes interprets a preamble (I’ve come to realise how important it is to get a preamble as precise / correct / unambiguous as possible but still as concise as possible). I also get plenty of feedback along the lines of “ooh, you’ve made that quite difficult, haven’t you, are you sure??”, but I mostly turn a blind eye to this because I’m aiming for quite difficult (quite difficult, just not impossibly difficult).
I certainly made more than a few amendments based on the feedback before submitting a version to the Listener editors. I was still a bit dissatisfied with some aspects (like the thematic message in the preamble) and I could probably have polished some clues a bit further but it was time. And there it was, lodged in the pipeline for eventual consideration (that was May 2023).
The Listener editing pipeline (time length) is what it is, and in some ways I like that aspect of the publishing process. It means there’s a healthy population of setters, and hence a range of styles and difficulty levels in the puzzles published. And the editors are impressively thorough and accomplished. It does mean, though, that what’s fresh in the mind of the editors is no longer fresh in the mind of the setter, and some hurried memory-jogging ensues when the first considered feedback comes! For this puzzle that first feedback contained the slightly alarming sentence “.. when I came across a couple of significant issues ..”. Damn. I continued to read .. “.. I hope you may be able to rescue the puzzle.” Ah. All was not lost!
In a nutshell there was an issue with one of the omitted letters not quite tallying which was fair enough and a bit careless of me. However, the other problem which I hadn’t seen coming and which resulted in by far the greater amount of effort to remedy was with one of the entries. I’d initially used what I thought was a legitimate and common enough abbreviation in the grid, but unfortunately ISR (for Israel) isn’t in Chambers. That was a valuable lesson, and again careless. Of course not every answer has to be in Chambers, but I think with an abbreviation (already a concession) a setter may be asking too much if it’s not in the big red book. In itself this issue doesn’t look significant; it’s just a 3-letter word after all. However, I’d probably submitted to its use after exhausting my search for ways of avoiding it, so when it came to “the rescue” there was quite a ripple effect through other crossing entries and then beyond. Eventually I arrived at the final ICR (which is obviously still an abbreviation but which is in Chambers, although I bet most people would think of ISR = Israel, and have no clue what ICR is). Hence quite a number of new grid entries and new clues, but thankfully rescuable.
I was able to do a bit of clue-pruning as well in this re-iteration because the puzzle was pushing the limits in terms of space. One slightly irritating aspect at this point was that I hadn’t been able to make a totally convincing preamble phrase; at this stage it was reading “give CRAIG A WONDERFUL COMPUTABLE ANSWER”. I mean, who’s Craig?
In the final edit, and I have to thank the editors for this part of the rescue, we managed to get to WONDERFUL COMPUTABLE ALGEBRAIC ANSWER which was much more satisfying, and didn’t require a lot of final adjustments to clues. Like other aspects of the puzzle, it wasn’t essential to engineer this expression, but it was always within touching distance so worth the effort to get there. Clearly much neater than a meaningless alphabetical list of omitted letters, and better than the mysterious CRAIG! It was intended to allude to both Heron’s formula, which some solvers may have found via Wikipedia (or even known already), and also the route of calculating the answer by the Pythagorean triangle difference. Algebra either way. Whether it was wonderful depends on the eye of the beholder I guess.
So, there you have it, that was Fisher Transformation. I hope it was mainly challenging and entertaining. Well done if you completed it, and well done if you’ve made it to the end of this blog too! I’ve got a couple of other puzzles in the pipeline, so hopefully Mr Snaky will be back at some point.