8 by 13 – a hopefully Quite Interesting setter’s blog
Posted by clanca1234 on 6 June 2012
8 by 13 – A Setter’s Blog
The Fibonacci sequence has intrigued me for as long as I can remember. Especially fascinating is the Golden Ratio (obtained by taking a term in the Fibonacci sequence and dividing by the previous term), and its many appearances throughout nature. It first crossed my mind that it would be a good topic for a puzzle when I solved a mathematical puzzle in the Magpie, Petals by Elap from Magpie 30 in May 2005, whereby a line had to be drawn through every number in the grid, each number in the grid being part of a term in the Fibonacci sequence, culminating with a central zero. At that time I had not yet set a single puzzle, however, and the idea sat in my mind for a few years.
In late 2007, by which time the first Samuel puzzles had been published, I was watching the television programme QI, and happened to notice that there was a rectangle and spiral pattern as part of the set design – a rectangle and spiral pattern that looked strangely familiar. For those who’ve never seen the programme, early editions included this pattern overlaid on a big screen:
And later editions had this pattern incorporated into the desks at which the host, Stephen Fry, and the contestants sat:
Yes, the design was the Fibonacci spiral, obtained by arranging squares with lengths according to the Fibonacci sequence into a rectangle, and then drawing a circular arc to connect opposite corners of each square, as follows:
Immediately, the idea for a puzzle was born, whereby the final grid ended up with the solver drawing a Fibonacci spiral. At that point, and before letting my mind fly off to start going further into the idea, I sat down to work out what size grid this could involve. It was quickly apparent that there were only really two choices – a grid that was 13×8 in size, or one that was 21×13.
The next idea that sparked in my brain was to do with the phrase ‘21×13’. I’d been solving the Listener regularly since early 2005, and one of my favourite puzzles to date (and still one of my favourites today) was the fabulous 20 by 10, Listener 3852 from November 2005. This involved a large grid (20×10, surprisingly) in which the solver had to find the true title of the puzzle, and the setter’s pseudonym, in the completed grid. This stuck in my mind for two reasons – firstly, I really, really enjoyed this part of the 2005 puzzle; secondly, it took me over a week of hunting to find it. Whilst I had no desire to set solvers a challenge that involved them spending seven days getting increasingly grumpy to the point that their other half threatened divorce (!), I decided that I might try to utilise the same idea here.
Whilst it had been the ‘21×13’ grid size that had sparked this idea, it didn’t take long to realise that a 13×8 grid would be more manageable. Whilst I thought that a 21×13 puzzle might be publishable, the grid would surely have to be squashed in order to manage this, and it would be important for the validity of the Fibonacci Spiral that the cells in the grid remained square. So, 13×8 it was. My next step was to prepare an empty grid in Excel, as follows:
Assuming that I kept this orientation (the long sides of the grid being horizontal), then there could be four possible ways in which the rectangles could be arranged (and thus the spiral drawn). Which one should I use? I didn’t pursue that thought at that point, and instead wondered how solvers could draw the spiral. The first idea was to have all cells along the path of the spiral spell something, but it quickly became apparent that this would put huge constraints on the grid fill. Also, the arcs would, by their very nature, pass through many fragments of cells. If we were drawing straight lines that would be okay, but the arcs made that unworkable.
My next idea was clashes, which ended up in the final version of the puzzle. If the arcs had to be drawn through opposite corners of the six squares in the grid, this involved ten cells. What thematic word could there be that had ten letters? After a few seconds where I excitedly thought ‘the word FIBONACCI has ten letters’, I wrote the name down, counted, and found only 9. D’oh! That wasn’t going to work. At this point I paused for about a month, whilst I tried to decide whether I should go with just the nine clashes, make ten clashes by using L(eonardo) FIBONACCI, or come up with something else altogether. Nothing else dawned on me, I wasn’t keen on the untidiness of E FIBONACCI, so I went with simply FIBONACCI, as follows:
This also included SPIRAL SAMUEL in cells obtained by increasing Fibonacci terms. Taking 0 as being the starting point, +1 = the top left cell, +1 again = the cell to the right of this, and so on. This decided once and for all the orientation of the grid as any other orientation resulted in at least one clash between the letters of FIBONACCI and of SPIRAL SAMUEL. Now I could get a fill… or could I? There were still two things to consider; firstly, how solvers would know to divide the grid into squares (and how they would know the size of the squares), and, secondly, whether losing letters in the cells containing the letters of FIBONACCI should contain anything thematic.
After some thought, I decided to have a message from clues, which would indicate the division required. ONE ONE TWO THREE FIVE EIGHT (giving side lengths) + SQUARES was a thirty letter message, so it seemed obvious to try to have 30 clues for the puzzle. I’ve also always tried to use thematic clueing gimmicks for my Listener submissions (or at least gimmicks with some relation to the theme), so I decided that solvers would have to remove letters from clues before solving, the number of letters removed being the relevant terms of the Fibonacci sequence – ie 1 from the first clue, 1 from the next clue, 2 from the next, then 3, 5, 8, and then back to 1 again. This would give 5 cycles of the removals (ie 1, 1, 2, 3, 5, 8 letters x 5 times), so it seemed it would be tidy. This put a further constraint on a potential fill, but I was still optimistic of achieving this.
I was also optimistic that solvers would realise that the removals from clues were not simply arbitrary, but related to the theme – and perhaps provide another route to discovering the theme. Early feedback tells me that solvers did not always spot this, however.
Looking at the constraints for the grid fill, as in the diagram above, it soon became apparent to me that, even if I could think of something for the discarded clashing letters to spell, I would perhaps be lucky to get a fill at all. Particularly unhelpful was the top left hand corner, where the only words starting with the letters SP?IC were variants on SPLICE. The vertically positioned R, L and S to the right of this were also problematic, as these would need to be unchecked. Filling manually, because of the clashes, I first arrived at the following:
At this point, I was particularly concerned with the unching in SHADCHAN at 1dn, as this was effectively S?A???A?, which wasn’t great. I took stock, and worked on the fill on and off, always manually, and cursing that the positioning of the letters of SPIRAL SAMUEL weren’t more friendly. With this, and the concentration of unches in the left hand side of the grid, things were all rather tricky.
Eventually, six months after first coming up with the idea, and three months after being at the point shown in the diagram above, the final fill was in place:
It was now time to start writing clues. The gimmick I’d chosen made things pretty tricky, even though some fell into place quite nicely. This took a month or so, and then I was ready to send the puzzle out for test solving…
Except I couldn’t. The puzzle being one where the setter is anonymous, I could not think of a sensible way of sending this to a friend to test, without them knowing it was me (or at least knowing that it was one of only a few people). I thus decided to send it directly to the first vetter (back in 2008 this was John Grimshaw). This was also fraught with difficulty, as it had to be sent anonymously so that the vetter would have a fair solving experience. I set up a new email address (firstname.lastname@example.org), created submission documents in a format completely different to my normal ones, and sent it off. And waited. And waited.
Within a few months, two things happened to cause difficulties. Firstly, the puzzle 10 by 8 appeared. This was another puzzle with an anonymous setter (not me!), and concerned me as my puzzle, at the time of submission, was called 13 by 8. Secondly, quite a long time later, John Grimshaw handed over the baton of first vetter to Roger Phillips, amidst rumours of a sizeable backlog of puzzles. The second vetter, Derek Arthur, told me verbally that another of my puzzles was appearing shortly, but I couldn’t ask about 13 by 8 without revealing the setter’s identity. I waited for what seemed a very long time, and eventually received some reassurance from Derek Arthur, via my anonymous email address, that the puzzle had been ‘found’. We agreed some small preamble changes and a few clue rewrites – the most notable of which was the definition for TUBED. This originally had a definition of ‘travelled from East India to Stratford’, which required amendment as being too obscure, East India and Stratford both being London Underground stations.
Derek also made the change of the puzzle’s title from 13 by 8 to 8 by 13, on two counts – first/ly, 8 by 13 was more accurate mathematically (eg as in the classification of the size of matrices), and secondly, to avoid any confusion with the now quite old 10 by 8 puzzle.
Only a very short while after agreeing these changes with Derek, he sadly passed away. This was a very sad blow to me, as I am sure it was to many other setters. He had always been unfailingly encouraging and positive in his dealings with me, especially when I asked daft questions (of which there had been quite a few). During the vetting process for this puzzle, Derek told me a lot more about Signor Fibonacci (as he called him in our correspondence) and his work, for which I am also grateful.
Fast forward some time again, and it was a very strange feeling when the puzzle appeared. It was odd not having been able to mention the puzzle’s existence to friends prior to publication, for obvious reasons, and so I was more nervous than normal when awaiting feedback for the puzzle. As I write, this has so far been positive, although I appreciate that the definition for ARIF may have caused some solvers trouble, given the absence of the ‘some first names’ appendix in the current primary reference, Chambers Dictionary (2011). When the puzzle was set, the 2005 edition was the primary reference!