## Listener No 4501: *Two Solutions* by Quinapalus

Posted by Steve Tregidgo on 25 May 2018

This terrific puzzle provided a good example of how a Listener solver can use a technique unavailable in many cryptics: using the endgame to help solve the clues. In theory one could solve everything then use the answers/entries, misprints and so on to work out the final endgame solution. In practice the fact that the endgame exists means we can make extra deductions about the clues.

I always start with tally marks. This puzzle had 15 clues with two solutions and 32 misprints (with no overlaps), so I made some quick charts: if I reached a point where I had all doubles or all misprints, I’d know right away what form the remaining clues took. I also made counters for the abbreviations and gaps; by the very end it helped me remember I still had an abbreviation (EASDAQ) to find.

I also tend to build a table of all answers, for those cases where the entries differ. (For puzzles where entries need letters changing, it’s immensely helpful to have a record of the original answers when checking over one’s work.) In this case, ER__CAL from the corrections implied VERTICAL, and the arrangement of double answers suggested they’d be concentrated on that 38/41 line so the bottom of this table would be mostly doubles. That wouldn’t leave room for many nouns (such as LINES) and so VERTICALLY made sense. Without yet knowing how that would be used later, I could deduce some real definitions that had so far eluded me (“state leaVer”, “digital sTore”, “in which one’s nanny’s alleLes fit” and “romans held swaY”). Similarly when the GSON/ROLL combinations became apparent in 41 and I dredged the Charles Dodgson / Lewis Carroll connection from my memory, I could make much more headway knowing that, say, 39 two letters (EX from “Exchange”) were to go after V or W.

But where were those gaps? ARGAND DIA[GR]AM (with its sneakily new way of using a double-letter cell!) and the grid’s initial thick lines like axes (plural of axis, not of axe) told me that the endgame would be a complex plane: a way to visualise complex numbers, which are combinations of real and imaginary (square root of -ve number) parts. I found Carroll’s Phantasmagoria and its equation, “xx + 7x + 53 = 11/3”, determined the roots (using an online solver since I knew it would be complex) and could see the exact cells they corresponded to. That in turn told me that 1a and 46 had the gaps. At this point I should have also realised I was going to have to MOVE BOTH XS but I was missing a bit of that instruction; if I had it I’d have had two free Xs to fill in as well. Which makes me wonder: was a version of this puzzle considered where the cells above *and* below each X were considered as gaps?

Since this post is about using the endgame to help with the clues, I won’t dwell on the grid construction, but I’ll say that I very much enjoyed working out where the entries lived. It’s almost a complement to a jigsaw crossword; with a little logic, half of the (blank) entries could be marked immediately, and more could go in as the letters appeared.

Thanks Quinapalus for one of my favourite Listener puzzles this year.

## Steve said

Addendum: having looked at the scanned grids on the other blog posts, and remembering my own submission, I fear I’ve not been accurate enough with the vertical positions of my moved Xs (drawing pretty treasure-map style Xs in the boxes which look closer to +/-6.5 than 6.1). Drat. I assumed answers would always have to be in the usual places in their boxes, which perhaps is a silly assumption for The Listener.

## Dave Hennings said

Steve, I suspect you will be marked wrong, just as I will be! Be that as it may, I thoroughly enjoyed reading your thoughts on solving the puzzle.

## Steve said

Thanks Dave, that makes me feel a little better!

## Encota said

Steve, An interesting read – many thanks!

## Steve said

Thanks. I must say, your DJ / Dre theory was one I started wondering about (although I didn’t find the extra clinching evidence that you ferreted out).