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Listener 4354: Taxi! by Ilver

Posted by Jaguar on 1 August 2015

Oh dear, another long break — but somewhat enforced, perhaps, as I have remained incredibly busy of late. Managing to keep my Listener head just above water, though, and I thought I would surface properly for this one. After all, if a puzzle that appeals to my mathematical background doesn’t prompt me to blog, what will?

Ilver is another of that group of new setters appearing in the last ten years or so, whose first Listener came out just in time for me to have a crack at it (my solving career starting in 2011). Since then, his nine puzzles in all outlets have included themes such as Pig Latin, Doctor Who and Poirot in the Listener, but also a couple of puzzles that have had mathematical elements. As it turned out, this effort belonged to the latter category — the first hint of that emerging when those extra letters revealed something looking like “a very interesting number”. Oh, that quote! I’ve known about it for a while, GH Hardy telling his sick mathematical colleague, the brilliant Srinivasa Ramanujan, about the taxi he had come in (rather than, say, hoping he’s getting better and asking after the family?), and how boring the number 1729 is, and hoping that this isn’t some omen. “But no!” says the young Indian, “for 1729 is the smallest number that can be expressed as the sum of two [positive] cubes in two distinct ways!”

Of course, all numbers are interesting really (there is a ‘proof’ of this, because any smallest uninteresting number would be remarkable for just that fact, and therefore interesting again, and hence there can’t be an uninteresting number!).

At the other end of the puzzle, we are instructed to highlight an identity. And there is some of it, eventually emerging from those odd clue lengths, as “plus one equals zero” emerges below an “i pi”, and all that is missing from Euler’s identity is the initial “e”.

At this point, though, progress came to a juddering halt for a time. 1729 is the key to obtaining a second quote, and then something in the grid that is also related, and fixing the grid will insert that final e in its rightful place. But how to use the key remained a mystery to me (and, it seems, quite a few others) for some time! Perhaps it’s partly because the obviousness of the final highlighting provides something of a distraction. Is there a direct link between Hardy’s work and Euler’s? Much searching of their respective output follows, but isn’t too revealing. Nor does any quote about Euler seem particularly helpful.

But then, of course, this is primarily a word puzzle, rather than a mathematical one, so thoughts should turn to interpreting things a bit more cryptically. It also helps to check other GH Hardy quotes, of which one alone made it into my edition of ODQ. “Beauty is the first test: there is no permanent place in the world for ugly mathematics,” he said as part of a book filled with other gems. Interpreting 1729 as four separate numbers 1,7, 2, 9, and then taking the 1st, 7th, etc letters of across clues revealed the first part of that quote after all. Phew! (after much grid-staring and head-scratching).

To get to the second part, though? Another bit of grid-staring, although at least with the final E in mind thoughts turn to the 1st, 2nd, 7th, 9th columns of the grid (not rows, because we need that e in the second column/ sixth row). Towards the bottom of the 9th column is the Y of “Hardy”, a bit further above is the H, and the rest emerges if you take 1st/7th and 2nd/9th letters in alternating rows. Follow it up and an anagram of “mathematics” is sandwiched between “the world”. And rearrange that to its proper form and you see the e fall in its rightful place, to complete arguably the pinnacle of what is meant by mathematical beauty. “e to the power of i times pi plus one equals zero, where e is the base of natural logarithms, i is the square root of minus one, and pi is the ratio of the circumference of a circle to its diameter”.

What makes it so beautiful is what this identity brings together. In the first place, the numbers above are fundamental to mathematics in a way few other numbers are. And then the operations (exponentiation, multiplication, addition) are also fundamental to all arithmetic, each being used exactly once, as is the equality relation. There’s a certain elegance in everything being used exactly once, too.

But also, it is where these numbers come from that contains the real beauty. pi is a number from the classical Greek mathematics of geometry (and then Trigonometry, its offshoot developed in the Arab world in the 7th-9th Centuries). i has its origins in algebra, developed mainly in the 12th-14th centuries. And the number e emerges naturally from Newton and Leibniz’s development of calculus in the 17th Century. Even 0 is quite special, representing the abstract concept that took a long time to develop, that you can even quantify and work with nothingness. We can thank the Indian mathematicians for that one.

Thus it is, then, that the entire history of mathematics from the Ancient Greeks up to Euler himself can be contained and summarised in just five numbers and four operations. There can never be a purer and more beautiful result.

4354

 

 

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Listener No 4354: Taxi! by Ilver

Posted by Dave Hennings on 31 July 2015

This week, Ilver’s back. His last Listener was his puzzle celebrating the 50th anniversary of Doctor Who. This week seemed to have another television theme with the American sitcom Taxi… or not! Whichever, we had down clues with wordplay leading to an extra letter. After that, there would be a quotation, a key, another quotation, a contradiction, a corection that would resolve the contradiction and finally an identity to be highlighted — in lower case, of course.

Listener 4354There was nothing about the across clues, which was vaguely worrying as there would normally be an “Across clues are normal” sentence. It was also a bit of a concern that there were only two cells where the left and right sides of the grid linked.

1ac Beats time bars (6) looked as though it could be TRACKS or TRAILS. I decided to find out which and switched to the downs. 1 was alternate letters of tusk on a Thai ship for TSOT[A]SI and 2 was R[V]ED ANT. 3dn was, of course, another surprise — while it was given as (11) in the clue, there were in fact only eight spaces! 4dn was KETA and that confirmed 1ac as TRACKS

A few more clues came, including 14ac Roadway Model T car with old lady aboard (6) with its nicely misleading surface reading for TARMAC. 13ac To humiliate oneself brew tea with cold rag (7, two words), had me toying with CAGE RAT and RAGE ACT before finally settling for EAT CROW. That gave me CHAMPIONESS for 3dn, and the ONE could become 1 in the grid and then the PI could become Π. 10dn was PERIPLUS (8) for a five space entry, so PLUS became +. All this looked as though some sort of mathematical shenanigans was in front of us.

The northeast corner was a bit trickier, but the bottom right was fairly straightforward, with 19dn IMPISHLY enabling most of that quadrant to come together quickly. In the lower left quadrant I had a strange bit of luck. With 30dn Baroque composer cycles to one old gypsy woman (4), I got CHAI ([B]ACH cycled + I) and then got interrupted. On returning, I absent-mindedly cycled CHAI to give AICH and looked that up to see aich whow “see whow“. Looking that up, it gave me the answer to 11ac EH WHOW, which I hadn’t solved yet and, bizarrely, didn’t have its own entry in Chambers!

By this time, I had most of the extra wordplay letters in the downs and could see that they would eventually spell out A very interesting number. I knew that this referred to 1729, the number of the taxi that Hardy had taken on visiting Srinivasa Ramanujan (although I’d forgotten the names of the two characters involved).

With the grid completed, it was time for the endgame. 1729 was the key used to find the first part of a quotation. It didn’t take long to write 172917291729… alongside the across clues, and taking the appropriate letters in turn gave Beauty is the first test. This turned out to be from the same mathematician, Godfrey Hardy, and continues “there is no permanent place in the world for ugly mathematics”.

Moving on, we had to find a contradiction to this latter part and its author using the same key. I tried the same treatment on the down clues as the acoss, and got TNOCBACRIFIMW…. That was gibberish. How about working backwards from the end of the across clues: SUASGYLCARIM…, more gibberish. I tried working backwards in reverse clue order for both acrosses and downs to no avail.

Now although we were told to use the same key, it may not be used in the same way. Perhaps 1729 would become 1/7/29 or 17/29 or 17/2/9. More letter counting, and I could see a brick wall and failure looming.

At the back of my mind, I had been toying with the “(24 cells in total)” given by the preamble. There were 20 across clues and 22 downs, so it really seemed unlikely that they would provide the next step — yet it didn’t stop me wasting a couple of hours trying. The grid, on the other hand, was 12×12 and was a more likely source. The 1st letter T then move on 7 squares in the first row for O, then 2 A then 9 N then EETRART in boustrophedon fashion. Nothing. The same for the downs — nothing.

And then the letters from square 1 T, square 7 H (row 1), square 2 E — THE — that looked promising. Continuing with square 9 W, square 1 O and so on through every row. I was presented with THEWORAHSTTMMAEICLDHARDY. HARDY at the end meant that I was home and dry and in between THE WOR and LD was an anagram of MATHEMATICS. Thus the apparent contradiction of the second part was that we were presented with an “ugly mathematics” (ie anagram) in “the world” and that needed to be corrected. I originally thought that the contradiction would be some sort of quotation and from a different author.

Of course, the π + 1 = 0 had been evident in the grid all along, and with the new letters in place, it could now be expanded to give e + 1 = 0. This is Euler’s Identity and is considered by many (most) mathematicians to be the most beautiful mathematical equation using, as it does, five “special” constants.

Listener 4354 My EntryI suspect that the second part of this endgame may have caused a few problems, but luckily I’m not (I think) one of them. So thanks to Ilver for another excellent puzzle and a chance to investigate lots of other mathematical fun and games along the way (including 1+2+3+4+…=-1/12!).
 

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Taxi! by Ilver

Posted by shirleycurran on 31 July 2015

Taxi 1729 001Ilver! Renowned in the Magpie for the D and E level ‘Child’s Play’ crosswords that his children have put together. What are we in for? A first glance at the grid reveals almost two grids with just two gaps in that vertical line that separates them. There is going to be something fairly dramatic in the grid for that to get past the editors. Otherwise all seems to be normal, just a couple of open lights and a fine average word-length. I wonder what Ilver is up to!

Of course, we check the answer lengths and spot that there are four clues that seem to require answers longer than the space allowed in the grid – but we can’t solve any of those clues, so store the information for later.

Naturally I check Ilver’s membership of the Listener Setters’ Oenophile Club and am initially disconcerted to find ‘To humiliate oneself brew tea with cold rag (7 two words)’ [TEA C* + ROW = EAT CROW] That doesn’t sound like Ilver. Reading on, I come to ‘Essential acid is left in Edmund’s underwear (6)’ [IS< in LYNE = LYSINE – it’s an essential amino-acid isn’t it] – he must be on a health kick. I’m into the down clues before I find ‘Book stay here – exotic drinks (8, two words)’ [B + STA(Y) HERE* giving HERB TEAS] – clearly Ilver is on some sort of tea trip!

A very interesting number!

A very interesting number!

These clues are tough indeed. We struggle with the down clues until a familiar phrase seems to be emerging from the extra letters and the other Numpty tells me about the Hardy Ramanujan number – how Prof. Hardy went to visit the dying Srinivasa Ramanujan in his hospital bed, and, short of conversational openings, commented that the taxi he had come in had a very dull number. Ramanujan’s response is famous. “No, it id A VERY INTERESTING NUMBER; it is the smallest number expressible as the sum of two cubes in two different ways.” (I wonder how he would have coped with the three-monthly Listener numerical puzzles – ten minutes? It isn’t the fact that he was able to see that 1729 was the sum of those two cubes in two different ways, so much as the fact that he was able to instantly state that there was no other smaller figure that fitted the requirement).

1729. Clearly that is the ‘key’ we are looking for and as our grid fills, we wonder how we are going to use it. I fiddle with the letters in the grid, taking the first, then the subsequent seventh, move on two, move on nine and get … TOAEHE. Not promising.

Taxi 001The other Numpty has more success, reading the first and seventh of the first line, then the second and ninth of the subsequent line and so on … and it’s rather like the monkeys typing Shakespeare “Look, this one has finally typed something coherent – ‘To be or not to bogzugigubélih8ui*ç%%…” However, with persistence, we find an intriguing HARDY at the end of the grid, so we are clearly on the right lines.

We have also spotted a rather strange version of Euler’s ‘identity’ Ti pi + 1 = 0 when we finally realized that those extra long clues could be resolved by fitting the PI of SNIPING into a single cell, using + as the conclusion of PERIPLUS, 1 for the ONE in CHAMPIONESS, and = as the start of EQUAL SIGN. Obviously the O of GONOCYTE can do double duty as a zero. But there is that odd T where we need an e.

Taxi 2 001We try applying the same method to the clues and Eureka! We discover another familiar quotation: ‘BEAUTY IS THE FIRST TEST’ and suddenly things fall into place. Hardy again! ‘Beauty is the first test and there is no permanent place in the world for ugly mathematics’.

What do we have in our grid? An anagram of MATHEMATICS (so ‘ugly mathematics’) and, as the preamble told us, its author, HARDY. Oh but this is brilliant; it is almost on the Hardy/Ramanujan level! We unscramble MATHEMATICS and, of course, Euler’s identity receives its missing e, which we are told to write in lower case. I have just a small doubt here. Instinct tells me to use the π sign and that is a lower case letter – but what will happen if I use lower case pi?

That slight doubt doesn’t detract from our admiration of a stunning compilation. Thank you, Ilver!

 

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Listener No 4353: A Ghost Story by KevGar

Posted by Dave Hennings on 24 July 2015

KevGar’s third Listener this week after No 4210, with the left-handed pianist Paul Wittgenstein as its theme, and No 4288, a Haydn symphony puzzle. Perhaps this week we would be treated to The Phantom of the Opera.

Listener 4353Some clues contained a misprint, and I guessed that “some” could be in the range of 3 to 30. 9 ASEAN, 11 COFIRE, 13 NEB and 15 RAIA got me off to a quick start, including one misprint. It seemed natural to try some of the intersecting downs, and 2 REBUKE, 8 SEA DOG, 10 NAIA and 11 CARGO were soon in the grid.

The north-east corner was soon set, including ONDING at 21, which was a new word to me. I found myself motoring around the grid in a clockwise fashion which, if you are an avid reader of these blogs, you will know is becoming quite common for me.

With the grid nearly finished, I had IRGILS and NEID as the misprints from the across and down clues respectively. OK, so Virgil’s Aeneid was going to be the reference work. This was one of the set books that I had for Latin O level, specifically Book VIII. The other was Caesar’s Gallic Wars, Book V. Are they still fresh in my mind? No. In fact they weren’t even then. I have to admit that I found it all a bit of a struggle.

Anyway, back to the present, and my grid was soon finished. It had taken a little under two hours. All that was left was to find the quotation, draw some lines and replace some letters. Opening my ODQ, I was surprised to see how many quotations there were for Virgil, over half of them for Aeneid. Of course, they were given in the original Latin together with the English translation, and that left the question of which language would be in the grid.

It took a couple of scans through before “Do not trust the horse, Trojans…” made me stop and think, with Equo ne credite, Teucri. Quidquid id est, timeo Danaos et dona ferentes in Latin. A few minutes more, and TIMEO DANAOS ET DONA FERENTES was traced in the grid in the shape of a horse, well more like a knight chess piece, actually.

Listener 4353 My EntryThe “ghosts in some form needing replacing” were easy to find in the centre of the grid, and TGSOSH was changed to GREEKS to give new words. It needed minimal googling to verify CARRE as a French word.

All done and dusted, thanks KevGar. A pretty easy puzzle but good fun.

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A Ghost Story by KevGar

Posted by shirleycurran on 24 July 2015

Timeo Danaos 001The preamble wasn’t too daunting and promised us something artistic at the end where we were going to replace ghosts by something more frightening. ‘some clues contain a misprint of one letter in the definition’. That ‘some’ is always mildly disconcerting (in this case it could be anywhere from two to forty-three) though they did reveal themselves fairly quickly and spell out Virgil’s Aeneid for us so that we had a stab at the theme long before we had a full grid.

But I am leaping ahead to that rather nice Trojan dog that my ten straight lines created (he does look a bit like a scotch terrier, head on, doesn’t he?). First I had to do my usual scan through the clues to check that KevGar still qualifies for the Listener topers’ club.

What do I find? ‘Drug’ in three of six consecutive clues; ‘A cold symptom observed returning before drug (5) (SEEN< + E = NEESE), ‘Adult swallowed up by hard drug — cry of dismay (3)’ (A in H H) and ‘Reprimand detective (mostly) found with drug — no fine (6)’ (REBU(s) + KE(f) with that ‘no fine’ meaning ‘endless’) and even JOINTS at 23dn in our grid. I find food; a ‘Young person eating wild animals around start of dinner: they’re made for di[V]ing (12)’ that produces SPRINGBOARDS and our first misprint. I find a ‘Fish starter of grouper with bits of cabbage (6)’ (G + RUNTS), ‘Fluffy Scotch eggs’ and ‘partner swapping’, and someone applying paint ‘topless’. What are setters coming to! – Standards are clearly dropping – but not a drop of alcohol.

Still, these are attractive and approachable clues (there should be a club for setters who have managed to include TSETSE in their grid – or maybe a more elitist club for the stars who have managed to avoid it, and all the other old chestnuts like PA, the Maori fort or their MERI, that ubiquitous ASTI, or the revolutionary CHE). However, KevGar’s TSETSE soon revealed its thematic function so he might be excused.

Yes, we had been solving for under an hour when VIRGIL appeared and which of us didn’t study Book II of the AENEID in our Latin classes (and which of us remembers any lines other than TIMEO DANAOS ET DONA FERENTES?) Amusingly, in the context of the current Greek problems, the Numpties were quoting that ‘fear’ of what the Greeks could do to each other earlier this week. I remembered something about ‘ut’ and ‘et’ clauses in Latin that we looked up:

‘Fear clauses take the construction of ut/nē + subjunctive. They are terrifying, because the meanings of ut and become reversed. In fear clauses, ut means “that not” and means “that.”Timeō veniat. I am afraid that he is coming! Timeō ut veniat. I am afraid that he is not coming!’ (I think the current Greek mess is a bit of an ‘ut’ situation!)

Trojan dog.

Trojan dog.

… but, of course, what we found when we checked our sources was that older versions of the ODQ have one slight difference from current ones and that, inexplicably, ‘ferentis’ appears in the place of ‘ferentes’. Chambers resolved it all and the other Numpty cooked supper while I hunted in the grid for what had to be a Trojan horse.

An endgame like this is right up my street. None of that grid-staring that solvers on the message board are currently complaining about in Elfman’s ‘Revelations of John’. A rather lean horse quickly emerged and it took only a minute to find ‘GHOSTS in some form’ hidden in there and replace them with GREEKS.

We thoroughly enjoyed this crossword with its lovely endgame.  Many thanks, KevGar.

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