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Kasparov has chosen 15 Nc3.

Barnet chess club top players have provided an "element management" paper on the above position to assist team member understanding.
The "element management" paper
Okay, now you should have "the force" behind you, and are ready for some analysis of Nc3:-

Background to 15 Nc3
Nc3 takes the knight on e2 to a more aggressive square where it attacks the d5 square. This is important, because for example, Kasparov can now play the move Be3 targeting our vulnerable b6 pawn, without us having the resource of ...Nd5 which would be the case if the knight was still on e2 :-

The case when the Knight was still on e2- Nd5 is a strong reply to Be3
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C) 15 Nc3
F) 15...Ne4?!!! (The Gavriel Gambit variation)
SOME INTERESTING ORIGINAL ANALYSIS HERE !!!!
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Daniel King writes (12/07/1999) in the posting forum:-
15...Nd4?

clearly isn't working because of 16 Be3 Nd5 17 Nxd5 Qxd5 18 Qd2!

as a number of people have pointed out.
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Daniel King writes (12/07/1999) in the posting forum:-
15...d5!? is very loose:

it is too early to expose Black's king.
Daniel King writes (12/07/1999) in the posting forum:-
The current favorite move is 15...Rd8,

but to me this looks extremely awkwardly placed; if Black had time to
regroup, all well and good, but...
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Daniel King writes (12/07/1999) in the posting forum:-

I think White's strongest move is 16 a3,

and we saw earlier how Black was running into trouble after 16...Rd8 17 Re1 Ke8 18 Bg5 followed by Rc1 and the b5 pawn is vulnerable and there are hits on Bl's queen.

Therefore, (hope you are still following this!) my suggestion is 16...Ne4.

If 17 Nxe4 Qxe4 18 Re1 Qc4 19 b3?! Qd4!

exchanges queens (that's what Black wants!)
And 17 Nd5

can be satisfactorily met by either 17...Bd4 or 17...Qd4.

I do not present this as conclusive analysis, just something
else to
consider. This feels a more 'natural' way to play the position than
others.
This is becoming very interesting!
Daniel King.
"
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After Be3, the World can play Ra6 as suggested by top British GM Speelman in his regular newspaper column.

Barnet chess club's secret super-computer resources analyze this position to test our tactical solidity:-
Kasparov can play the obvious looking Rc1, with an X-ray attack on our Queen:-

We can reply Ne4 trying to simplify things, and take the punch out of Kasparov's pressure:-

Kasparov can continue with Nxe4 Qxe4, Qb3:-

We can reply with Qe6:-

Here there are two options examined:-
20 Qb5 and
20 Rc4
20 Qb5

Here there is the tactical threat of RxN followed by QxR. Therefore the world would have to parry this threat somehow.
After Qxa2, the position is computer assessed as roughly equal:-

20 Rc4

You can trust a computer to come up with a weird looking move like this!
In this position, we have the sneaky resource Bxb2 if we want it. The thing to bear in mind are that White's queenside pressure is potentially increasing, and therefore we need to calculate carefully.
This fails to the simple 16 Nxe4 followed by 17 Qb3 winning a pawn

The "Gavriel gambit" variation:
We could play 17 Bd4!? and sacrifice the f7 pawn:-

For example:
After 18 Qxf7 Ne5 19 Qb3 we have the powerful centralising move 19...Nd3 :-

with interesting compensation!
For example
A) 20 Qa4+
Kd8 21 Be3 Nf4! :-

Here if Bxf4, then Bxf2+ winning the queen or if Bxd4 then Qxg2 mate.
Another variation:- After Ne5 Qb3, Nd3
B) 20 Qb5+
(instead of Qa4+) Kd8 21 Bd2 Rf8

with clear advantage to black!
C) Bg5

We have Nxf2! Rxf2 Bxf2+ Kxf2 Qf5+ Qf3 Qxg5 =
eg:-
24.Qxb7+ Ke6 25.Qxb6 [25.Re1+ Kf7 26.Qxb6 Qh4+ 27.Kf1 Rf8 clear advantage to black] 25...Rf8+ 26.Kg1 Qd2=
D) Be3?! exchange sacrifice
21.Be3 Bxe3 22.fxe3 Qxe3+ 23.Kh1 Nf2+ 24.Rxf2 Qxf2 25.Rf1 Qc5!
Alternatively if White plays 19 Qf4 instead of the retreat Qb3 then:

Simply 19..Qxf4 20 Bxf4 Ra8!

with a playable position!
British International Master Simon Ansell has verified the strength of the gambit variations above. There is an early divergence instead of the materialistice Qb3- there is Qd5.
eg:-
1.e4 c5 2.Nf3 d6 3.Bb5+ Bd7 4.Bxd7+ Qxd7 5.c4 Nc6 6.Nc3 Nf6 7.0–0 g6 8.d4 cxd4 9.Nxd4 Bg7 10.Nde2 Qe6 11.Nd5 Qxe4 12.Nc7+ Kd7 13.Nxa8 Qxc4 14.Nb6+ axb6 15.Nc3 Ne4 16.Nxe4
16.Qd5 Qxd5 17.Nxd5 Bd4 18.Be3 Bxb2
A) 19.Rab1 Be5 20.Rxb6
A1) 20...Rb8 21.Rc1
(21.Rfb1 Ke6 22.Nc7+ Kd7 23.Na6)
21...e6 22.Nf4 Nc3 23.Rc2;
A2) 20...Ke6 21.Rfb1;
B) 19.Nxb6+ 19...Ke6 20.Rab1 Be5]
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Diagrams will help illustrate a very important posting made by Tom Casanovas. He has used a range of top computers to analyse the position after Nc3. Despite some obvious move mistakes in his posting, there is some insight to be gained from examining briefly what the computers think the implications of Nc3 are.
But remember to use the force! Computers sometimes need to be turned off!!
He posted (Sun Jul 18 03:54:34):-
Thanks to the dozen of answers to my last mail ... although
some of them a bit too agressive or authoritative for my liking. Perhaps
we have lots of GM here? ;-)
I have decided to try to help the World Team putting six top programmes to work
simultaneously in Pentium Computers for about 96 hours in the position arising
from 15 Nc3. The number of position analyzed is around 2.000.000.000 each.
Please do not understimate the strength of such programmes. All of them
have at least 2.500 elo FIDE under Tournament conditions and one of them (Hiarcs
7.32) probably around 2.600 . Their conclusions are therefore of
relevance. Let's know their main lines of analysis:
A) Genius 6.01 doesn't like 15 ... b5. It doesn't include this option as
one of the three main lines and prefers
A1) 15 ... d5

16 Re1 e6
17 Bg5 Ra8
18 a3 Qg4

A2) 15 ... e6

A3) 15 ... Ra8
16 Be3 Ra6

17 Rc1 e6
18 a3 d5

In all the variants Genius seems to opt by the pawn structure e6 - d5 for
black. It considers white position slightly better.
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B) SHREDDER 3. The current World Champion (Padeborn 99) evaluates the position
as better for white both with 15... Ra8 and 15 ... b5.
B1) 15 ... b5

16 Re1 b4
17 Na4 Kc7
18 b3 Qd5
19 Bf4

B2) 15 ... Ra8

16 Re1 Ra5
17 Bd2 b5
18 Rc1 b4
19 Na4 Qd3
20 Nb6+

Shredder's strategy is the advance of b pawn to b5 and b4, either directly or
after the preparation Ra8. You will have seen that it is a bit pessimistic
about 'our' future in this game.
C) WCHESS 2000 preferences are for 15... b5, although it is also
pessimistic about black situation.
C1) 15 ... b5

16 Bg5 b4
17 Bf6 Bf6
18 Nd5 Bd4
19 Ne3 Qe6
20 Qe2 Be3
21 fe3 b3
22 a4 Kc7

C2 15 ... Ra8

16 Re1 Kc7
17 Bd2 Nd5
18 Nd5 Qd5

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D) CRAFTY 16.6 This very strong program made by Bob Hyatt (and
available for download for free!) thinks that White's advantage is minimal both
with 15... b5 and 15... Ra8 variants.
D1) 15 ... b5

16 Re1 b4
17 Na4 Qg4
18 Qd3 Qf5
19 Qc4 Ke8
20 Bf4 Kf8
21 Rad1 d5

D2) 15 ... Ra8

16 Re1 Ra5
17 a4 Re5
18 Re5 Ne5
19 Be3 Nd3
20 Qf3 Nb2
21 Rc1

The idea Ra8-a5-e5 seems interesting for black. By the way, Crafty's
suggestion after 15 ... Ra8, 16 a4 is 16 ... Ke8 with the idea Nd7 and Kf8 and
evaluation almost equal : + 0.12
E) ZARKOV 5 considers 15... Ra8 followed by 16... b5 as the best line for black,
leading to a position with a sligth advantage for white:
E1) 15 Nc3 Ra8
16 Re1 b5
17 a3 b4
18 Na4 Ra6
19 Bc3 Ke8
20 Nb6 Qb5

F) HIARCS 7.32 (IMHO the best commercial program in the world today) evaluates 15 b3 as an interesting option for white as an alternative to 15 Nc3
F1) 15 b3 Qh4
16 h3 Nh5
17 Rb1 f5
18 Bd2 Qe4
19 Rc1 Ra8
20 Rc4 Qd5
21
a4 evaluation
: + 0.60
Against 15 Nc3, Hiarcs 7.32 prefers 15... b5 with some advantage for
white:
F2) 15 Nc3 b5
16 Be3 Ng4
17 Nd5 Ne3
** The following MUST be incorrect- editor's note
18 fe3 e6
19 Qf3 f5
20 Nf4 Kd7
21 Rfc1 Qf7 evaluation :
+ 0.59
After 15... Ra8 16 a4, Hiarcs recommendation for black is
F3) 15 Nc3 Ra8
16 a4 Ne5
** The following MUST be incorrect- editor's note
17 Qd5 Qd5
18 Nd5 Ra6
19 Be3 Nc5 evaluation
: + 0.35
The white's line 16 a4 seems not as good as other alternatives
against 15 ... Ra8, for example 16 Re1.
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Barnet chess club top player conclusions
This is a critical point in the game. The world does not have an obvious best move here. The team of Barnet's top players are analysing the implications of Ra8, Rd8, b5, and Ne4?!! but unfortunately have no conclusive evidence of the best move here. Ra8-a6 was judged as interesting but funny looking. b5 looks natural and aggressive. Rd8 looks a bit too optimistic. Ne4 needs to be seriously analysed!
World, the decision is yours!
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Outcome for interest
Top 5 votes:
h8 to a8 - 48.15%
b6 to b5 - 14.71%
h8 to d8 - 10.99%
d6 to d5 - 4.93%
f6 to e4 - 4.22%
Ne4?!!! reflections by other Chess Sites
SmartChess online:-
GM Ron Henley's analysis and commentary
"In an impressive display of team-work, the World met the challenge of having to make a VERY difficult choice as it opted for 15...Ra8 (an original suggestion of English GM Jonathan Speelman). Late-breaking analysis on the World Team Strategy Bulletin Board demonstrated that 15...Rd8 was a good alternative (albeit a little unclear in some lines), and that even the at-first-glance dubious 15...Ne4 could not be so easily dismissed thanks to some fine analysis by Barnet Chess Club. Meanwhile, Irina's original suggestion of 15...b5 never quite recovered from the barrage of actual and attempted refutations and dangerous transpositions launched at it on the World Team Strategy Bulletin Board over a two-week period - sufficient to dissuade the young analyst from persevering with this sharp and dangerous idea. The risky 15...d5 never survived the clean refutation that hit the World Team Strategy Bulletin Board shortly after it was revealed that the move was being actively examined."