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Sep2010
  Once: Hmmm ... an interesting final position. White threatens Qxg5 and then Qxh6+ or Rxh6+. But Fritzie finds a way for black to prolong the game. 24...d5 opens up the fifth rank for the black queen to join in the action. Now 25. Qxg5 is met by 25...Qxf6. White now has to win by picking up pawns because black seems to have enough to defend the mate. Two possible variations: 25. Nexd5 Bxf6 26. Nxf6 Kg7
click for larger viewWhite's advantage is less than a pawn.
25. Nfxd5 Kg8 26. Rc1
click for larger viewWhite has won one pawn the black c pawn will fall soon. But there's no immediate mate in the offing. Fritzie says +2.22 

Sep2010
  al wazir: <ruelas007: the pun has something to do with integrals? I´m just learning Integral Calulus> <Chessgames.com> isn't a good place to learn calculus. It isn't a very good place to learn about puns either. According to the bio (Nigel J Kalton), Kalton "was a distinguished mathematician" who died three weeks ago. 

Sep2010   nuwanda:
has somebody an idea about the pun "Converges to Infinity"?at first sight this sounds like a contradiction. where is the connection the game and/or the players? 

Sep2010
  gars: As a teacher of Mathematics I would say that "converges to infinity" is a contradiction in its terms, because if something goes to infinity it diverges by definition. 

Sep2010   goodevans: It's a very long time since I did my maths degree, but I would more readily recognise the term <Converges at infinity> to <Converges to infinity>. Many sequences and series converge as they approach infinity and for those like <ruelas007> who are studying integral calculus, both Lebesgue integration and Riemann integration are based on this idea. 

Sep2010   goodevans: P.S. Simple example:
The series 1 + 1/2 + 1/4 + 1/8 + ...
converges to 2 as the number of terms in the series approaches infinity. Colloquially we might say this "converges at infinity", but that would be slightly sloppy language. 

Sep2010
  johnlspouge: < <goodevans> wrote: [snip] Colloquially we might say this "converges at infinity", but that would be slightly sloppy language. > I agree. The phrase is usually redundant, because a sequence cannot converge any other way. The phrase "converge to infinity" is perfectly acceptable to me, however. A sequence can diverge to infinity (rather than converging to a finite number), but it can also converge to infinity (rather than oscillating). Nowadays, with the human capacity for analogy, I just think of infinity as an extended real number. Recall General MacArthur's quote:
"Old soldiers never die, they just fade away."
At a New Brunswick summer science camp I had the pleasure of attending in 1970, we teenage protoscientists made up several analogs, of which I still remember two: "Old chemists never die, they just fail to react."
"Old mathematicians never die, they just tend to infinity." < <al wazir> wrote: [snip] According to the bio (Nigel J Kalton), Kalton "was a distinguished mathematician" who died three weeks ago. > Between <al>'s kibitz and mine, the intent of today's pun should be quite clear. 

Sep2010
  kevin86: Black resigns in the face of having his game ripped to shreds. 

Sep2010   goodevans: <kevin86: Black resigns in the face of having his game ripped to shreds.> More specifically, there is no reasonable way of stopping <Qxg5> followed by <Rxh6+>. <24 ... Bxh6+> doesn't help because of <25 Qxf6+ Kh7 26 Nf5>. 

Sep2010
  desiobu: <Once> nice find with 24...d5 

Sep2010   rapidcitychess: I looked at this game and at several points my "quick idea" is that black is looking good. But as soon as the knight lands on f6, black's game looks suspect. First off, the Bishop (Archer, Runner, Laufer) looked like it had to be traded for that horrible knight on f6. Before that, the knight shouldn't have gone to f4. Once you get an knight on Q6, make sure your opponent doesn't get one on KB6. Rapidcitychess
^^ \/ 

Sep2010
  JohnBoy: I knew Kalton as s mathematician. I wish I knew at the time that he was a chess player as well. 

Sep2010   scormus: Yes, W is clearly better but I did think the 10 was a bit premature. <coincidentally .....> 1964, Dulwich School .... Nigel Kalton would have been a senior. Wasn't there another keen player at Dulwich, about that time? Who later became became rather distinguished in the chessplaying community. And who occasionally posts here ;) 

Sep2010
  chessgames.com: For the record, Daniel Freeman objected to the pun "Converging to Infinity" for academic reasons, noting what's been already said: series that don't converge at a finite values are said to DIVERGE, not converge. (At least, that's what they teach freshman calculus students.) But we're not a math site; we're just trying to come up with snappy game titlesboth "Diverging to Infinity" and "Converging at a Finite Value" sounded both humorless and perhaps even a tad morbid. 

Sep2010   rapidcitychess: <chessgames.com>
I find this all funny. Maybe we should make more controversial puns. :) 

Sep2010   pferd: It may not conform to any accepted mathematical definition, but one could say that a series "converges to infinity" iff the reciprocal of its sum converges to zero. Certainly not every divergent series has this property so the term is not trivial. 

Sep2010
  acirce: Good thing chessgames.com isn't a Swedish site. 

Sep2010
  watwinc: Am I missing something, <once>? 25 Nfd5 K8 26 Qxg6 

Sep2010   fetonzio: after Qxg5 white threatens only Rxh6, not Qxh6 

Sep2110
  Once: watwinc: 24...d5 25. Nfxd5 Kg8 gets us to here:
click for larger viewNow 26. Qxg6 looks interesting because of the knight fork on e7. But let's run through it: 26. Qxg6 Qxg6 27. Ne7+ Kh7 28. Nxg6 Kxg6. White's advantage is now just a pawn or so. <fetonzio: after Qxg5 white threatens only Rxh6, not Qxh6> It depends on how black responds to white's threats: If black takes the Nf6 first we get this: 24...Bxf6 25. Qxf6 Kg8 26. Nf5 and Qg8# follows. No white piece captures on h6. Or black could try delaying the bishop recapture by one move: 24... Rab8 (or any other pass move) 25. Qxg5 Bxf6 26. Qxh6+ Kg8 27. Qh7# The white queen captures on h6. Or black could leave the bishop on g7 all the time. Then white would mate with: 24...Rab8 (a pass move) 25. Qxg5 Rb5 (another pass move) 26. Rxh6+ Bxh6 27. Qxh6#. The white rook captures on h6. The point is that white has so many attacking threats that black cannot defend them all. But for white to win, he has to be aware of the attacking possibilities of Qxh6, Rxh6 and the Qf6Nf5Qg8# combination. 

Sep2110   nuwanda:
the nice thing about a (infinite) series that converges according to the mathematical definition is, that this series then reprsents this number and you can calculate with it just as if where indeed this number. you loose this property if you allow (whatever the definition should be) "converges to infinity". you cannot calculate, in a classical way, with infinity, e.g. something like "infinity=infinity" is just rubbish. and so <johnlspouge>'s <The phrase "converge to infinity" is perfectly acceptable to me, however. A sequence can diverge to infinity (rather than converging to a finite number), but it can also converge to infinity (rather than oscillating). Nowadays, with the human capacity for analogy, I just think of infinity as an extended real number> doesnt make much sense from a mathematical point of view. 

Sep2110   goodevans: <nuwanda ... you cannot calculate, in a classical way, with infinity> Correct.
There is a branch of mathematics, transfinite set theory, which deals with the fact that there are many different infinites (in fact an infinite number of them!) and deals with how to calculate with them. However, as <nuwanda> states, the rules for calculation differ from the classical rules of arithmetic. 

Sep2110
  johnlspouge: < <nuwanda> wrote: [snip] you cannot calculate, in a classical way, with infinity [snip] > Indeed, much depends on your aim. I spend much of my time doing applied probability, where the existence of a limiting random variable is usually a short step on a longer road. Infinity can emerge as the limit if an unusual event occurs, but it causes no computational difficulties if it has probability 0. (In most probability calculations, because of the rules of Lebesgue integration, effectively 0 times infinity = 0.) One should also recall that "classical" calculation is not the only game in town. Asymptotic series usually do not converge; this does not stop them from being very useful in many practical numerical computations. 

Sep2110
  Once: Old golfers never die, they just lose their balls.
And perhaps old chessplayers never die either  they just reset the pieces for another game. 

Sep2110
  Phony Benoni: Old chessplayers never die, they simply become resigned. 


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