Has it really been over four years since Quinapalus’s last Listener?! The theme of that puzzle (No 4297 *German Serial Composition*) was the fabulous Hedy Lamarr and her contribution to mobile phone technology — frequency-hopping.

This week, a carte blanche faced us — well, blanche apart from barred lines dividing the grid into its four quadrants. Eventually, it would have mirror-symmetry about the horizontal axis, which was unusual. Fifteen clues contained two definitions to words differing by one letter and wordplay to the common letters. In the relevant cells, the two letters had to be entered diagonally “so that *all* grid entries can be read”. Another weird preamble-ism, and were the italics just to ensure that I read the word correctly?! The remaining clues had misprints in their definitions.

I started off reasonably well, with 7ac BERK, 8dn EMIGRE, 9dn RIZLA, 11ac TUXES (loved the PJ/DJ misprint), 13ac THIAMIN and 17ac SMUGGLE/SNUGGLE, my first non-misprint clue. I thought it’d be fairly safe to put most of these in the top right corner, but that didn’t mean that the remaining clues in that quadrant were a doddle. As time progressed, the bottom right corner turned out to be a bit of a pig as well.

Especially with a long preamble, it needs to be double-checked to make sure nothing is overlooked that helps the solving process. The first thing that I needed to reread the preamble to remind myself about: “Numbers in brackets are the number of cells in entries, two of which initially contain a gap.” Consequently, I failed for far too long to realise that 1ac and 46ac had only 3-letter answers, SET and SAE, despite their (4) designation. Other traps I set myself were thinking that the correct version of “wellie” at 7ac was “wallie” rather than “nellie”, and overlooking the 3-letter entry at 12dn by putting four bars at the top and bottom of column 4.

All in all, this led to the bottom left corner being a bit of a pig as well! Luckily, the letters in column 7 helped me see RHYME AND REASON. I’ve been doing these puzzles for too long not to look for what the alternatives could spell out, and PHANTASMAGORIA was there. Together, these two hints led me to the Lewis Carroll book *Rhyme? and Reason?*, which I thought had been the subject of a puzzle before, but the Crossword Database didn’t give anything. Come what may, CARROLL and DODGSON could go in at the end of row 11. The start of that row would obviously contain LEWIS, but how would CHARLES LUTWIDGE be entered? (In fact, it would be REV CL.)

I put the alternatives I had so far into row 11 and column 7. In fact, the options could go diagonally in either order. Were these then the two solutions referenced by the title, and would JEG have to accept them in either order? Oh well, his shoulders are broad.

Referring to the book, it didn’t take long to track down the riddle, being one of four in a section near the end:

Yet what are all such gaieties to me

Whose thoughts are full of indices and surds (At least that resolved the nellies/wallies problem.)

x² + 7x + 53

= 11⁄3.

This can’t be right, the quarterly mathematical wasn’t for a couple of weeks. However, that good old quaratic formula, which was drummed into me at school, came to the rescue!

x = (-b ± √(b² – 4ac)) / 2a

So, with 3x² + 21x + 148 = 0, x = (-21 ± √(441 – 1776)) / 6

Well that stumped me, as it required the square root of -1335.

Time to change tack for a bit. I still had to decipher 14dn which looked as though it ended in DIAGRAM. At this point, a grin crossed my face as I saw that this determined the order of the two surnames and thus the two book references. JEG would have a (relatively) easy time, after all.

Eventually, I found ARGAND in Chambers — “a gas- or oil-lamp admitting air to both the inside and outside of the flame” — but that was of no use at all! I needed to read the preamble yet again to remind myself that 14 wasn’t in Chambers but in the ODE:

“**Argand lamp** an oil or gas lamp fitted dwith a cylindrical burner… ” Drat! Back to that lamp again, but at the bottom of the previous column I saw:

“**Argand diagram** a diagram on which complex numers are represented geometrically using Cartesian axes, the horizontal axis representing the real part of the number and the vertical coordinate the complex part.”

Yet again, I resorted to my favourite mathematical site Wolfram Alpha and keyed in the equation. Unfortunately this time, it totally confused me, giving the result as:

+*i*/6 × (√1335 + 21*i*) or –*i*/6 × (√1335 – 21*i*)

I don’t think we did Argand Diagrams at school, and I’m not sure we even dealt with √-1 and *i*. Be that as it may, having jumped ahead to the endgame, I still needed to complete the grid and unravel the message spelt out by the corrections to misprints: **Indices surds. Move both x’s vertically**. (If nothing else, this helped me fully understand 4dn *What’s written after Is? They follow Hs in lexicon (4)* with *Hs* becoming *Xs* — psi words following chi words in a Greek dictionary, I’m assuming.)

The two X’s were in column 3 and moving them up/down into the empty cells gave SEXT and SAXE, leaving behind TUES and MIEN. Their main role though was to tell us where to plot the two values of x, but what exactly were they? Luckily, Wolfram came to the rescue again with a button to show “Approximate forms”. This showed x ≈ -3.5 ± 6.0896*i*. So I plotted a point half way along cell 3 in the top and bottom rows, and approximately 0.1 up/down.

At least, I *think* that’s what was required of us. Whether I got everything right, or made a silly mistake, this was a phenominal puzzle from Quinapalus. Many thanks for a fascinating journey.