From time to time during the 19 years I’ve been setting thematic cryptic crosswords, I’ve asked myself whether I should try creating a numerical puzzle. Apart from the need for a suitable idea for the theme, I had some concerns about ensuring that there would be a unique solution, and that it would be reachable by a sequence of logical steps with a minimum of trial-and-error calculation. My first introduction to numericals was in October 2019 with the Magpie puzzle “HexaFlex”, a base-16 puzzle Tim King (Encota) and I co-set under the pseudonym EP. For that, Tim was the driving force on the numerical side, in a puzzle where digits became letters on the faces of eight connected cubes. This gave me confidence in venturing into the world of numerical puzzles, with my first solo effort, “Pathfinder”, published in the Magpie in September 2022.

Having submitted that one, I immediately started work on another as I had a theme in mind which promised to provide a novel, and hopefully interesting, numerical puzzle. The theme was quantum mechanics, especially the concept of the collapse of superposed states. With its strangeness, for me this branch of physics has always had its own charm. My target was the Listener series.

From the start, the plan was a numerical of the type where integer values have to be assigned to a set of letters. The twist would be that each letter would exist in a superposition of “states” (values) and would “collapse” into one of them, depending on which clue was being “observed”. Moreover, two different letters might assume the same value if they were in different clues. Initial explorations using a superposition of three states per letter showed how intractable such a puzzle would be to set (never mind to solve!), even with some constraint placed on each letter, such as possible values being consecutive. So a rethink was needed.

What I arrived at was two consecutive values for each letter, with across and down clues each using consistent values, with no repetition of values. I felt this to be much fairer to the solver, and the intention wasn’t to set a puzzle of horrendous difficulty. But there’d still be plenty of scope for a solver to make a slip! At that point, the title “Even Odder” suggested itself, given the nature of the number pairs, and the fact that quantum mechanics is just that!

I decided that every clue would be a “formula” using a thematic word. With some help from online resources, I was able to assemble a list of nearly 100 words (excluding plurals etc.) relating to quantum theory and particle physics. At that point, I introduced the feature of digit-triple sums in the grid’s columns leading to some relevant wording below the grid, using A=1, B=2, etc. I thought of quantum physicists’ surnames, but that looked too obvious, and so I decided to use a paraphrased quotation. This gave me a feel for what the grid dimensions would be. The setting process then evolved into a non-linear one, with several requirements having to be met simultaneously. These included how many, and which, letters to use to give me a workable subset from my thematic word list, and that answer lengths were to be no more than 6 digits (to avoid solvers facing overlong calculations) yet maintaining a reasonable overall average. Also, each triplet in the grid would need to add up to a number in the range 1 to 26. My final choices were 16 letters representing 32 values (leaving around 70 thematic words to choose from) in a 10×9 grid with 48 clues, and no entry longer than 5 digits.

So after much head-scratching, and over 30 A4 pages of closely-written notes, I had a completed puzzle which I had some confidence in. Happily, my feelings were confirmed by my two, much appreciated, test-solvers — sincere thanks guys, you know who you are! — and I was able to submit the puzzle to the Listener.

In any numerical, there has to be at least one clue, or a combination of clues, acting as an “entry point” for the solver. I decided to use factorials for this purpose, allowing the answer for 11a to be directly entered in the grid as the only 5-digit factorial (8! = 40320). In conjunction with 12a, it was then possible to discover four letter values. Consideration of 8d, another “factorial clue”, allowed a further six values to be tied down. So ten found, twenty-two to go! The process could be readily continued until all letter assignments had been found, and the paraphrased quotation discovered. Other variations on the entry point exist, of course.

I’m very grateful to the two Listener vetters for their careful appraisal of the puzzle, including some tightening up of the preamble wording. All those solvers who commented on the puzzle, using whatever means, also have my thanks.

Phil Lloyd (**Ploy**)