L4621: ‘True’ by Piccadilly
Posted by Encota on 11 Sep 2020
Thanks first of all to Piccadilly for a beautifully constructed puzzle!
I still recall well the puzzle The Properties Of Numbers II from three years ago, so knew we’d be getting something good. I wasn’t disappointed!
Of course the error-checking facility provided by the number-to-letter conversion at the end removed one of the most common gripes heard about Numericals: “I posted it off but have no idea if it was correct”, or similar.
With hindsight I should have made more use of the fact that no cell would contain a number greater than 19, but I kept wondering if there was some kind of trap in that logic (e.g. there might have been a 61 in the down clue and a 16 in the across clue). Luckily that proved not to be the case!
The way in with this puzzle (for me at least) was to start with the powers/ exponents used in some of the clues, e.g. 29d’s T^A. even if T=2 then A can’t be more than 9. Similarly the R^R in 21a limited R tightly.
I maintained a simple table in Excel (above), showing letters in one direction and their possible values (1 through 19) in the other. As one clue or another allowed certain values to be eliminated, I marked that clue number in relevant cells in the table. I always find this useful if (when!) I make a mistake somewhere along the way. [Aside: if you use Excel but don’t use Cntl-Enter yet – for filling multiple cells with the same value – then it can save you much time in tables such as this].
All great fun – many thanks!
Tim / Encota
Listen With Others 
Steve Tregidgo said
I was sorely tempted to go the spreadsheet route but plumped for pencil and paper, which I found satisfying. I set up a table just like yours, but with an extra row above the top one to make it easier to look up letters from numbers (which would prove useful near the end when back-filling some entries based on the emerging message, which in turn gave more letter values). But I just crossed out invalid items — I hadn’t thought to record an invalidating clue! That’s a great idea and will definitely help in future when I want to reassess my logic. (I did write one more list, the letter/number pairs in order of discovery, which allowed me the option of partially resetting to a more certain point in the solve — but it wouldn’t have let me restore my table.)
The other thing I was tempted to do (as I am every time a numeric of this sort comes up) is write a more generic solver — not to cheat with of course, but as a fun exercise. For example, I might set up a Python script (or a spreadsheet for instant-ish responsiveness?) which includes all the expressions and lengths, and can highlight when I’ve got letter values which can’t work. The next step from there would likely be to pick out expressions which have only one or two unknowns, and try all the permutations to see what can be filtered out (which is something I often do by hand to whittle down the possibilities). Next would be telling the script how entries interact (that this entry crosses that entry — harder in the case of the entries here!), and finally I’d want to replace the letters with pseudo-numbers that exist in a sort of virtual superposition of possible values. By this point I’d have a fairly complex constraint solver so I imagine the effort involve would far surpass that of solving by hand — but as I said, it would be a fun exercise…