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Listener No 4503: Property Management by Smudge

Posted by Steve Tregidgo on 8 June 2018

I don’t write blog posts for every Listener; I tend to wait until I have a tip or a solving technique to share. These tips may well be obvious to many, but maybe some solvers will have not thought of them, and posts like this will help them next time.

This ingenious numeric was most easily solved on two grids: the one provided, and my own table mapping grid numbers to rules. I started by marking out a 40×19 grid: one row for each grid number (with extra rows for 10a/10d and 23a/23d), and one column for each rule (I wrote the rule letter and the count at the top of each column). Much of the solve was spent eliminating possibilities in this grid.

To start with, and to serve as an example on a smaller scale, there are only six squares in the range 1-38 (1, 4, 9, 16, 25, 36), three cubes (1, 8, 27), two fourth powers (1, 16) and two fifth powers (1, 32). The only one of those categories where the required count matches the possibilities is the fourth powers, so we have 1 and 16. With the 1 and 16 spoken for, the squares must be 4, 9, 25 and 36, and the fifth power must be 32 (and there’s only one three-digit fifth power, so that gives us our first entry too!). That just leaves the cubes to decide between, and the very next rule takes care of that: the only possibilities are 1^1=1, 2^2=4 and 3^3=27. 1 and 4 are accounted for, 27 is the only remaining option, and so the cube must be 8.

I therefore spent half of my time crossing off boxes in my table where I knew the grid number didn’t match the rule. By the end of that process there would often be the correct number of viable choices in the rule column, or just one choice in the grid-number row, so another grid-number could be matched with a rule (I drew a circle to confirm the match) and more possibilities eliminated. This was a fairly simple process. The other half of the time was spent trying to eliminate rules based on the grid entries. For example, primorials of 2 or more digits must always end in 0 (they have factors of 2 and 5); primes of 2 or more digits are always odd (they cannot have a factor of 2). So any grid entry with a final digit can immediately be eliminated from one of those rules.

No discussion of a property-based numeric is complete without mentioning the single most useful resource for solving them: the Online Encyclopedia of Integer Sequences. Here you will find all squares, cubes, fourth and fifth powers, primes (Mersenne, Fermat or otherwise), triangular and tetrahedral numbers, factorials, primorials, palintiples (rule ‘s’, where the multiplier is not 1)… the list goes on. You can find a sequence by name, or by typing in a few of its members. And from any page, the “links” section usually has a table of several hundred or thousand entries to pore over.

For example, there are fourteen fourth powers with 6 digits, giving a relatively short list of possibilities for 16d. Six of those have zeroes that would become the first digit of a crossing entry in 1a, so that leaves just eight possibilities there. When I have a list that short, I write them all out for further elimination later. My final tip is that squared paper is helpful here: it becomes a lot easier to read down the columns, especially to discover that the Nth column only ever contains particular digits (which can feed into a crossing entry).


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Listener No 4503: Property Management by Smudge

Posted by Dave Hennings on 8 June 2018

Well, this quarter I was actually geared up for this quarter’s mathematical puzzle — so often they catch me by surprise. However, I wasn’t geared up for the setter. Having expected Nod or Zag to make a reappearance, I was faced with Smudge, and he didn’t ring any bells regarding a previous puzzle. I was surprised, therefore, to find that he had set one of the tough puzzles of 2016 over two years previously — No 4388 Cycle 20% More, all about Gilbert and Sullivan’s The Grand Duke, not a mathematical at all.

This week, we had a series of properties in a list, such as square, cube, triangular number, Fermat prime. Each grid entry had to be associated with exactly one of these properties, as did the clue number at which it was entered. We were also provided with information about numbers divisible by their reverse, perfect numbers and integers which are the sum of two squares, all of which “may be helpful”.

I loved the clueing technique here, similar in many ways to Piccadilly’s The Properties of Numbers — II last year. I made a list of the clue numbers and then went through annotating each to show which of the given properties applied to them. I gradually teased out some grid entries but, two hours later, I reached a dead end — my first. This was, I think, because I had put 56 as a definite for 10dn, rather than just a possible.

I decided to be a bit more organised on my second attempt, and created a grid with clues down the left and properties/occurrences across the top:

This made it much easier to tick off the entries as I resolved them or to mark properties that didn’t apply to a clue.

I also used my favourite mathematical tool, WolframAlpha, to identify the following:

  • tetrahedral numbers, n(n + 1)(n + 2)/6: 1, 4, 10, 20, 35, 56, etc
  • Mersenne primes: 3, 7, 31, 127, 8191, etc
  • Fermat primes: 3, 5, 17, 257, 65537, etc
  • Perfect numbers: 6, 28, 496, 8128
  • Numbers whose reverse is divisible by the number: 1089, 2178 and palindromes

Unfortunately, none of this prevented me from diving headlong towards dead-end number two, which was overlooking 901 as a possible entry for 34ac (reverse of a prime but not a prime).

Third time lucky, and my grid looked like this:

And I managed to successfully complete the puzzle like so:

Thanks for an excellent mathematical, Smudge.

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Property Management by Smudge

Posted by shirleycurran on 8 June 2018

Preamble almost as long as the clues and not a hint of alcohol in either – well, what did I expect? Certainly not the two days of almost reaching a full grid then spotting a problem, a badly chosen fourth power that would lead us up a blind alley or a missing rule i from our final countdown. Yes, I know that success depended on keeping a very careful record of which rule we had used and which were still available for use but I wonder how many solvers sailed straight to the solution in a few well-documented minutes.

Friends sometimes say “Aren’t you lucky that both of you are interested and solve crosswords together!” If only they knew! The other Numpty is the mathematician so I initially got out of hearing range weeding the bottom of the garden, but eventually had to participate in the grid-filling and record keeping and that was fraught and even ended in the odd shouting match. The air was blue at times.

We were frustrated by the need to consult lists of Mersenne primes, triangular numbers, five-digit primes and so on. Is there somewhere a solver who has these at his finger tips? We certainly don’t and the heap of paper piled up.

Just a portion of the mountain of paper this produced.

Well, we got there in the end and are agreed about one thing – what bliss that there isn’t another of these for three months. Can it really be true that there are more entrants to these things than to the verbal ones? Where do all those batty solvers come from?

Thank you anyway, Smudge. At least you gave us an alternative to watching little children dressed as Union Jacks and waving the same at the royal event. The other Numpty claims it was a very ingenious construction.

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‘Never-ending?’ by KevGar

Posted by Encota on 1 June 2018

I love Treasure Hunts, especially those involving puzzle-solving!  You may know the type, where you have to unlock something before another feature will work.

Question. But first, how could a puzzle be Never-ending, I hear you ask?

Answer. Why not put something sneaky in it – like that cryptic many years ago where about five clues were normal but all the rest contained crosswordese-sounding rubbish (you may recall what I mean, such as “Reservists confused by top-class uprising“, that sort of thing), leaving solvers perplexed and unable to reach its end?  Now that’d be Never-ending!

In Listener-speak (i.e. theme-based puzzles) this could also happen nearer the puzzle’s overall completion – we could perhaps refer to it as the NeverEndGame – where it appears there’s a way to complete it but actually there isn’t.  If one was to do this, it would be only be fair if one gave a hint that something bizarre was happening – maybe a synonym for ‘bizarre’ on Row 1, for example?  But wait a MINUTE, what’s this I see at the top of the Grid below?

2018-05-12 16.40.44

In this week’s puzzle there were ten hidden words in clues and ten synonyms for them appearing  in the Grid.  For me the ten were (in no specific order):

  • 16a Female – SHE 6d
  • 10a Marijuana – POT 32d
  • 19a Sister – CLARE 23a
  • 35a Urine – PEE 33d
  • 36a Later – AFTER 26a
  • 15d Port* – RIO 13a   *more later…
  • 1d Elves – PERIS 32a
  • 9d Referee – HEAR 12a
  • 18d House – QUINTA 11d
  • 24d Foot – PES 33a

Twenty other letters were hidden in twenty clues, as an extra letter in each wordplay.  After a while it became clear these spelt out six words:


I didn’t recognise it but soon looked it up and found the phrase


associated with the Danish composer Carl Nielsen.  And there he was (after a couple of minor changes, to an R and an E), hiding on the trailing diagonal in the grid.  So far, so good.  The Preamble even told us that these letters (R & E) would be the first and last of one of the ten left-hand words in the Bulleted list above and there, sure enough, was RefereE.  That only leaves nine of the words! Excellent!!

The Preamble then told us to combine a list of ten letters (obtained elsewhere in the puzzle) with the first or last of the other nine words on the left-hand side above – FemalE, MarijuanA etc.  The ten letters obtained elsewhere were: A,E,T,P,S,T,H,T,N,O.  So, using [AB] to mean A or B (but not both), we had to find an appropriate work by Nielsen that used, in any order, 19 characters from:


Now, you might think that is where the answer would just pop out but no, this is The Listener, that’d be far too easy!  One might at this stage have checked the completion date of Nielsen’s ‘Inextinguishable’ 4th Symphony (1916, if you are asking), and then spend hours scanning the Titles of all his earlier works (there are lots, trust me 😉  ) for a 19-letter title using the above.  You might also initially have been quickly drawn to his 2nd Symphony – DE FIRE TEMPERAMENTER, or (in English) THE FOUR TEMPERAMENTS – but, on closer inspection would find it wasn’t quite all present in the letters above.  Darn!  So you’d then scan all his earlier songs, and then his earlier string quartets for obscure subtitles and then repeat all the above in any other languages you felt might be relevant – but to no avail.

So the NeverEndGame would appear to have succeeded in achieving its (lack of) end.  Now what?

But, wait a moment, let’s have a closer look at
A,E,T,P,S,T,H,T,N,O,[FE],[MA],[SR],[UE],[LR],[PT],[ES],[HE],[FT].  [Really? Ed.]

If you carefully untangle this one gets:


Clearly that was the Hidden Message all along!  Apparently the first solver to email the Editor with this phrase wins – what a fantastic Treasure Hunt!  [Well, that’s my interpretation of what happened, anyway]  And only on receipt of such an email does an Automated ‘correction’ get issued.  So, soon after, this appeared on the Listener website:

Listener No 4502 Correction

A correction has been added to Listener No 4502 as follows:

CORRECTION: In Listener 4502, the clue for 15dn should read “Tree debarked and chopped down around university metropolis filled up again”.

A new PDF will be made available as soon as possible. We apologise for the error.

And in the Bulleted list above, ‘Port’ was now changed by this to ‘Metropolis’ and the possible pair [MS] replaced [PT] in the search above, now becoming
Once that had been unlocked/established, the Treasure Hunt was easy to complete and Nielsen’s Symphony #2 ‘The Four Temperaments‘ at long last came into view.  And, fortunately for us successful solvers, the QM in the Title, ‘Never-Ending?’ was justified after all.



Tim /Encota


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Never -ending? by KevGar

Posted by shirleycurran on 1 June 2018

After muttering about an over-long preamble with too many things going on, the other Numpty started solving at such a rate that all I could do was wield the pen and try to keep up. Clearly music was a theme as Rachmaninov appeared in the first clue – even if he only gave us an R for ur anagram,’After bizarre opening of Rachmaninov music is later orchestrated (12)’ SURREALISTIC, with an extra M, declared the Numpty, and followed that with SYNTHESISERS at the opposite end of the grid – but we had an extra word there: LATER – that was going to give us a potential L or R and define a word that was to lose a letter. An original device!

I hardly had time to scan the clues to check KevGar’s retention of his place at the bar but there was plenty of evidence as our solve progressed. ‘Sneaks back in service as beer brewing comes around (9)’ We couldn’t work out the wordplay of that one but REABSORBS fitted our grid and brewing BEER seemed to give us an extra E. ‘Scary actor using old rum dropping dead (6)’ was less of a problem. We dropped the D(ead) and got LUGOSI with the anagram USING OL giving us an extra N. ‘Tree debarked and chopped down when imbibing top class port tanked up (9)’ added port to the beer and rum – no wonder he was ‘tanked up’! Cheers KevGar!

We decided that the PORT was extra in that one and defined RIO which had lost a T in clue 13. (t)RE(e) was FELLED around U, so REFUELLED. That was another fine, generous clue and we were keeping a careful record of the MARIJUANA/POT, FEMALE/SHE, SISTER/CLARE, URINE/PEE, LATER/AFTER, ELVES/PERIS, REFEREE/HEAR, PORT/RIO, HOUSE/QUINTA and FOOT/PES that were going to give us a choice of first or last letter to unscramble together with the ten letters docked from defined words. Those, we decided, were TTTSHPANEP.

It was CARL NIELSEN who appeared next, along one of the diagonals (where else!) and we used the two letters of REFEREE to correct the spelling of his names so we knew what 19 letters we had to produce the work that preceded the INEXTINGUISHABLE. ‘Music is life and like it ….’ our extra letters had told us. There was the infinity symbol coiled round the centre of the grid and Google told us that we were looking for THE FOUR TEMPERAMENTS to write below the grid. Still, it is always a good idea to check — and consternation: our letters didn’t work. We needed another M and we seemed to have an extra P. It sounds as though KevGar has invented temperapents – some sort of intermediate ski slopes. Thank you anyway!

Later: Ah, a correction has appeared that gives me a metropolis instead of a port so goodbye to my intermediate ski slopes!

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