Eclogue's puzzles tend to stem from a certain cause and effect. A conversation here, a dvd watched, a book read and "Decline and Ball", our latest puzzle in the Magpie series is no exception.

In May 2011, Logogriph won the Enigma puzzle (entitled 'Perplexing Patio') in the Prospect magazine for which the prize is a recently published book with a mathematical bent. On this occasion , the book in question was "Perfect Rigour" by Masha Gessen, which narrates the background, build-up and aftermath of Grigori Perelman's proof of the Poincare Conjecture, one of seven Millennium problems for which the Clay Mathematics Institute offered a $1 million dollar reward for verifiable proofs. To date, it is the only one to be proven. Like Andrew Wiles' proof of Fermat's Last Theorem - this is the sort of work that really should be far more widely known (though not necessarily understood), but will hopefully have had an engaging appeal with the Magpie fraternity.

The conjecture states:

As Wikipedia helpfully describes:

"For compact 2-dimensional surfaces without boundary, if every loop can be continuously tightened to a point, then the surface is topologically homeomorphic to a 2-sphere (usually just called a sphere). The Poincaré conjecture asserts that the same is true for 3-dimensional spaces"

There have been various crosswords in recent times on mathematical feats or impossibilities (Mash's Listener Salver winner based on the Klein Bottle, being the most recent that springs to mind), so this seemed appropriate fodder, but needed to be reduced to a more accessible form. The idea of using a circular diagram as a torus to illustrate the general conjecture was always going to be a good starting point, as was adding the complexity of running lights around the grid entry. Subsequently, a similar mechanism was used in the construction of "Seasons Greetings IV by Eclogue" which appeared on the Crossword Centre website in December 2012.

Logogriph had trouble with the variety of possible published spellings for Mr Perelman's given name, before settling on using the Anglicised equivalent of Gregory. This was never designed to be a mathematically tortuous exercise, but one which marked an extraordinary feat of cerebral dexterity and of boundary pushing by a complex character growing up through an ever-changing Communist regime.

At heart, Eclogue can only admire the achievement and the principles which Grigoriy Perelman (building on the Ricci flow work of Richard Hamilton) showed in refusing the $1m dollar prize fund which went with the proof.

Even for the numero-phobe, the "Perfect Rigour" book is an excellent read, not least for the extremely positive portrayal of maths teaching in communist Russia, which demonstrates the great importance placed upon the subject as well as the merits of stretching the most able students.

Our puzzle was undoubtedly far simpler than that faced by Hamilton and Perelman, but hopefully no less 'beautiful' or 'enjoyable'. "Decline and Ball" appeared in Issue 123 of Magpie published in March 2013 as a Grade D (the toughest grade thus far given to an Eclogue puzzle by Magpie).

Magpie's Homage to Decline and Ball |

We were delighted to receive the following responses from the Magpie audience:-

- I like circular grids but I got poincare early and wikipedia povided the other two rings so it didn't take long to complete. A very nicely constructed grid.
- After reading the preamble I thought this puzzle would be about Perelman and the Poincaré conjecture, and I was glad that I was right. I had enough problems with getting help from the checking letters, and I think it would have been quite a bit harder without having knowledge of the theme to help place the answers.
- Having only solved a few clues I worked out which letters of POINCARE occurred in each answer and this was enough to determine how his name would appear in the central ring. After I had a few more answers I tried to work out where things were going in the other rings. GREGORY PERELMAN and CLAY MATHEMATICS INSTITUTE were obvious, but ONE MILLION DOLLARS and HAMILTON less so.
- MILLENNIUM PROBLEMS was pretty easy to see in the clues and then I needed some sort of MORPHISM. I can see that the Wikipedia article says that Hamilton's work involved spaces diffeomorphic to spheres, but I would say at best this property is involved in the answer rather than the question.
- D grade looks right. I ran out of time on this, but thoroughly enjoyed the work I put in.
- I've managed to solve 13 clues only: not enough to get a foothold in the grid. I can see a likely ...OMORPHISM and, at the end, PROBLEMS coming but no help from that.
- No: the notorious Listener torus puzzle left scars.
- This was one of those dam-burst puzzles, little feedback from the grid initially, before the messages made things much easier. From a personal point of view I would dispute that Tarot cards reveal anything. Also I did wonder why "yoof" rather than youth? These aside this was a fine treatment of an all but intractable theme.
- Luckily I knew of Perelman and the Poincare Conjecture beforehand: I suspect, once again, that less mathematically inclined solvers will have been at a huge disadvantage here. For me, very satisfying and a perfect justification for the circular/toroidal grid.
- Lovely treatment of an amazing theme - quite a story. Why did he not accept the $1 million and give half to Hamilton?
- Very difficult clues with highly obscure definitions and well-concealed misprints. I may have made an error - or misunderstood a clue; I do not understand how WARMER clues MEANER.
- The dreaded 'viewed as a torus' in the preamble - I can never solve torus problems - I think it's a spatial awareness issue. But hopefully that is a thing of the past having completed this puzzle. I still don't understand the overall concept (and thankfully don't have to) but it was a fascinating subject. The circular grid suited the theme and Eclogue employed a similar trick to their recent Crossword Centre puzzle which I actually found much harder. The biggest surprise in this puzzle was discovering someone turned down $1,000,000 in prize money.
- Surely much better than the month's Magpie puzzle, and certainly more interesting; but what a subject to inflict upon the general reader ... Diffeomorphism, indeed! I had heard that the Conjecture had been proved, but had forgotten and am glad to be reminded.
- Brilliant puzzle. I was more familiar with the theme than many solvers will be, I guess, although 'diffeomorphism' was new to me.
- Took a while for the penny to drop, largely because having solved 25,27,28 and 30 early, I had .H.MI.T.., which I led me to CHEMISTRY ! and a complete red herring. Then thinking about the preamble led me first to Hilbert's problems and eventually to the Poincare conjecture, and that quickly led to all the ring answers. Even so I had to carefully count the different clue types to finally solve 32., which of course had very simple wordplay (for an unfamiliar word)
- Didn't think that this rated a D. More like a B.
- Took a long time to get going - I HATE MISPRINTS! From the 10 answers I had, I worked out DIFFEOMORPHISM and PROBLEM. After some time googling TORUS I stumbled across PERELMAN, HAMILTON and POINCARE. With a couple of guesses to complete the rings I then set about solving all the other clues.
- It all seems to fit but there's a couple of clues that I don't fully understand. However, my answers do fit the entry constraint of four groups of 8.
- A definite D.
- I like circular puzzles but the entry method and grid meant that this was almost entirely a cold solve - indeed even finishing it off was a slow process of considering a lot of options. I wonder whether the third ring from the centre couldnt have been shared between two clue answers.
- Another inspired guess - declined ?mathematical prizes - after a slow cold solve of 14 clues cracked this one open.
- Very enjoyable torus puzzle which took less time to solve than Two Codes by Wan but which was definitely harder to break into. I started at radials 13,14, 16, 17, 18 19 and 20 and worked outwards until deducing millenium and diffeomorphism which speeded up the rest of the solve. PANGEN was my last entry. Very enjoyable, thanks Eclogue.
- Great puzzle. I thought it was unnecessarily cruel to avoid giving the lengths of entries only half of which were known for sure at the outset.

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