# Archive for May, 2018

## Listener No 4501, Two Solutions: A Setter’s Blog by Quinapalus

Posted by Listen With Others on 27 May 2018

I first encountered the riddle used in Two Solutions in one of the books in Martin Gardner’s excellent Mathematical Diversions series. Its rhythm had stuck in my mind, and I had been thinking for a while that it might be possible to use it as the basis for a puzzle. Also, I’d been experimenting with ways to make it easier to construct grids where lights have two solutions in Qxw — with the current version it is possible but fiddly — and thought this might make a good test case.

I could see that the quadratic equation would have two distinct solutions, and on determining that they were complex, the idea of using the grid as an Argand diagram came naturally enough. Given that the coefficients were, I assume, chosen purely for metrical reasons it was good fortune that the roots can be plotted in a reasonable-sized grid with the side of a cell representing one unit. It was also fortunate that the real part of the roots is an odd multiple of one half so that they lie in the middle of a square horizontally, which makes the grid neater and the entries easier to check. In combination with the word ‘vertically’ in the instruction message it also gave some solvers some confidence that they were on the right track.

The complex roots of a polynomial with real coefficients always come in conjugate pairs: that is, pairs differing only in the sign of the imaginary part. This is because if you replace i with –i throughout a polynomial equation it will still be true; or as someone put it ‘how do you know that i is the positive square root of –1?’. It was therefore thematic for the grid to have up-down mirror symmetry. To avoid ambiguity it was important that there were no Xs in the grid other than those marking the roots; with a bit of work it also proved possible to make the grid pangrammatic.

Many solvers seem to like puzzles that have a little maths dust sprinkled on them, though I’m sure there are some—not too many I hope—for whom the endgame came as a slightly unpleasant surprise. To them I apologise (but only a bit). For the mathematical puzzles in the Listener series the editors use GCSE level as the reference for an acceptable level of difficulty. (Presumably if the same threshold were applied to the expected level of familiarity with, for example, English literature, setters would only be allowed to draw themes from Macbeth.) As far as I can establish this puzzle probably only just slips below the GCSE bar. Complex numbers are taught at that level, and so is the method of ‘completing the square’ for solving quadratics, which is convenient to use in this case as the coefficient of x² is 1. What is less clear is whether the combination of these two ideas falls within the syllabus.

It is unfortunate that ‘Argand diagram’ is not defined in Chambers but on the other hand there is an excellent explanation in Collins and of course in many online sources. All in all I was a bit apprehensive when I submitted the puzzle that the editors might reject it as demanding too much mathematics but in the end they let me get away with it. Nevertheless, to be on the safe side I arranged to be out of the country when the puzzle was published.

My thanks to the test solvers and the vetters for all the improvements they suggested, to the checker, and to the many solvers who gave feedback either online or along with their postal entries. Thankfully the response was mostly positive and there was no angry posse lying in wait for me at Heathrow. Your comments are all greatly appreciated.

Quinapalus.

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## Listener No 4501: Two Solutions by Quinapalus

Posted by Steve Tregidgo on 25 May 2018

A portion of my working table, shortly before finishing.

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.

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## Two Solutions by Quinapalus

Posted by shirleycurran on 25 May 2018

We are staying at the bottom of a cliff on Ibiza with relatively limited solving resources and had to drive across the island to the nearest ‘locutorio’ and cope in Ibizenco (the local version of Catalan) to get our weekly challenge – and the moment that I saw a quadruple carte blanche and the name Quinapalus, I knew that it was going to be that challenge. I read through the preamble and gave an awed gasp but did, of course, confirm that Quinapalus reserves his place at the Listener bar: after all, he is a previous winner of the Ascot Gold Cup. ‘Ale is recognised by these kings quaffing one (6)’ gave us SHAHS who were rather surprisingly drinking that beer but we put one (I) into them and got SHIAHS. So cheers, Quinapalus.

The preamble warned us that we were looking for someone with two preoccupations that would appear to us just before a riddle in a work of one or three words in yet another unclued light. Some cells were going to contain two letters entered diagonally – obviously leading to two different answers and there were going to be two gaps. There was one redeeming feature – top-bottom mirror symmetry.

We solved rather slowly as it was our turn to cook (for ten!) and these clues had the Sabre touch but with a few assumptions, I managed to begin a grid fill with that obvious KERB* giving us a Nellie or a BERK, and AITU, EMIGRE, RIZLA and KNEEPAD intersecting with it at the top right. Word lengths provided a putative grid but we were rather baffled by 38/41 and a string of clues from 30 to 42 that seemed to lead to those double solutions with letters entered diagonally: WITLESS/WITNESS, SCLIMS/SCRIMS, POOTER/POSTER, SCREE/SIREE, GLARY/GOARY, VEX/WEX (a new word for me and somewhat appropriate as the Ibiza temperature had just dropped to 8 degrees and it was pouring while we heard that in London the temperature was 29 degrees!) SUNG/LUNG and DORT/RORT.

We should have found LEWIS CARROLL and the REV C L DODGSON a lot earlier – an obvious choice of theme for Quinapalus with his mathematical riddle. ARGAND. DIAGRAM appeared using the diagonal GR of the two names but it took us several more hours and some Googling to find the remaining diagonal letters that helped us spell out PHANTASMAGORIA and RHYME AND REASON. I found the first riddle and groaned to see that I had to solve a quadratic equation and that somehow the solutions were to be marked in the ARGAND DIAGRAM obviously using the axes of the original carte blanche grid.

INDICES SURDS, MOVE BOTH Xs VERTICALLY, the corrected misprints told us. Fortunately there were five rather high-ranking CERN physicists in our party and I passed my problem over to them. X2 + 7X + 53 = 11/3. ” (Solving a quadratic equation is like falling off a log, they told me – it’s simple.) That’s minus 3.5 and plus or minus the square root of 37 i – that’s just a tad more than 6″. Then ensued a long discussion about exactly where in the grid we had to put those two Xs that we were moving vertically and marking (reasonably) accurately in the grid. I think that word ‘reasonably’ is an editorial addition to avoid another Poat hare. We have to have two solutions so those Xs must move just over the demarcation of their original cells (in TUXES and MIXEN) so that the also complete the words SEXT and SAXE and fill the empty cells and we have our two solutions. Too clever for me! It took us twenty-four hours to solve. I wonder how long it took to set. Many thanks to Quinapalus.

## ‘Two Solutions’ (or ‘Hip-Hop ft. Dr Dre’) by Quinapalus

Posted by Encota on 25 May 2018

I don’t know about you, but I found this one hard!  I am not a particularly fast solver but this one was much nearer to 10 hours than the (perhaps) more typical 3+ hours-ish!  I finished at around 1 a.m. Sunday.

The puzzle contained two clue types.  I’d seen this first clue type once before, I think, and even had a go at writing one or two in the past.  To recall, the clue was in a form such as:

<Definition1, with N letters> +

<Definition2, with N letters> +

<Wordplay for the (N-1) letters that Defs 1 & 2 have in common>

A gentle example appeared at 30d:

See stupid confuse wisest (7)

‘See’ is WITNESS, ‘stupid’ gives WITLESS and the rest of the clue ‘confuse wisest’ is the jumble-based wordplay for WIT.ESS.  Simple, eh?

The other clue type featured a misprint in each definition which had to be removed before solving.  Many of these I found tough, for example:

Raised letter containing leak and joint buckled (7)

I could see from checked letters very early on that this must be KNEEPAD – but why?

At long last I spotted that buckled should become BUCKLER, a protector according to Chambers, then it was PEE ‘N’ in DAK ( a letter), all reversed.  Always hard when two of the pieces of a clue are unknown to you!

Once a few letters started appearing in 5d I had .H.N..ND…… and wondered if it might be CHANSON DE GESTE, the Song Of Roland etc but that soon didn’t fit with other crossers.

[surreal mode on]

OPTION 1:
What with DJs appearing in various stages at 26ac and, later on, at 11ac, plus the mention of rapper Dr. Dré himself in 29ac, the theme was obvious:  hip-hop.  Given the entry at 5d, it was clearly all based around Peter Spirer’s 1997 hip-hop film documentary Rhyme and Reason  https://www.imdb.com/title/tt0120014/

This was soon confirmed by finding, in the grid, RAP, in Row 5,  (Dr.) DRE at 29ac, and the hip-hop stars NAS and ICE-T from Spirer’s film in contiguous cells in the grid.  And, what with HIP and HOP being only one letter apart, this was surely part of the solution.
OPTION 2:
There are those who feel I have completely lost the plot – and that the Solutions referred to in the Title are actually all drinks.  Again, finding in contiguous cells, LAGER, RED, RUM, TEA & CHA, as well as (Creme de) CASSIS already at 35ac gives some substance to this argument.  And when I found that PHANTASMAGORIA is a cocktail – involving melon & raspberry liqueurs with pineapple juice, if you’re asking – then that pretty much proved it.  Maybe.  But whoever heard of Listener setters & solvers liking alcohol, or a cup of tea, or both?  I therefore obviously discounted this option.
OPTION 3:
Finally, I’m riddled with some vague thought that a Quadratic might be involved somehow … but as that’s only of second order, I’ve ignored it.
[surreal mode off]
We’ve had some amazing puzzles in Listener 2018 series already but this must surely be one of the finest.  Quinapalus has taken school-level Maths combined with a famous Oxford don’s poetry to create one of those delightful crossover puzzles that should suit polymaths everywhere.  Tough wordplay, an astounding grid, multiple angles to a theme including poetry, solving quadratics and the oxymoron that is simple complex numbers.  Though I suspect not everyone will agree, an ideal Listener puzzle, in my humble opinion!
Cheers all,
Tim / Encota

## Listener No 4501: Two Solutions by Quinapalus

Posted by Dave Hennings on 25 May 2018

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 + 21i) or –i/6 × (√1335 – 21i)

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 ± 6.0896i. 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.

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