Characters that clearly like problems include the folks at Chess Cafe, Cornel Pacurar (who has a links page), Ian Gent, Vincent van der Bilt and Manolis Stratakis. Nandakumar Sankaran has an Endgame Studies as does IM Igor Khmelnitsky. Shaihin very kindly sent me not one but two studies laid out in beautiful LaTeX PostScript.
As ever, I wrote this piece for our juniors.]
Chess problems
"It is quite a threepipe problem".  SHERLOCK HOLMES"A chess problem is an exercise in pure mathematics"  G.H. HARDY
You may have often seen chess puzzles with the challenge of 'White to play and mate in two'. Well, the area of chess problems of this sort is really a whole extra chess continent, with strange customs and unusual wildlife. A wellformed problem is a composed position with a single, often surprising, solution. Many different sorts of problems exist, and most of them look as though they really have been composed  that is, nothing like that would happen in a real game.
Problem thinking is a bit different to chess thinking, but the same principles apply  once you've seen a pattern (theme) you can find it more easily in other settings.

SOLUTION
1. Kb1 b42. Bc1 b5
3.Rd2 Kf4
4. Rd4 mate
The problem is 'cooked' by 1. Bh1 or 1. Rd8, but the solution is essentially the same.
This I believe was very difficult to solve at the time, becuase it was a new type of problem; now, the "Indian theme" is wellknown to solvers.
Want to try some? Here is a threemover, a retrograde problem and a 'fairy chess' helpmate problem (helpmate = Black to play and help White mate!). They are all pretty tough for beginners, so you might just prefer to play over and enjoy the solutions rather than try to solve them.
In the Fairy Chess problem below, there is a new type of piece on the board at c7  the CamelHopper. This attacks the squares just one more square distant than a N does  that is, on b4,d4,f6 and f8. But it moves only by moving over an attacked piece and moving the same distance the other side  so, if the bQ moved to d4 white could play 1 CHe1, leaving the Qd4. If the bQ was on f8, CH couldn't moveto "i9", and in fact can't move in the diagrammed position. Clue: how to control b1?

(a) S.Loyd 1869
The Lovechase: Black invalidation: White nonpermission.: Mate in 3

(b) R Smullyan '57
Where is the wK?

ANSWERS
(a) 1 Qe4 fails to 1...g6. Sadly, 1 Ng6+ hxg6; 2 Qc8+ isn't fast enough, and 1 Qc8 (threat 2 Ng6++) fails to 1...h6. The retreat 1. Qf1! is the key  Now, if 1...Bb2; 2 Qb1, or if 1...Bc3 or 1...Bd4; 2 Qd3, or if 1...Be5 or 1...Bf6; 2 Qf5, or if 1...g6; 2 Qxa1#.
(b) K is on c3: the previous moves must have been: 1 Pc2c4, Pb4xc3e.p.+; 2 Kb3xc3+. Clever stuff! Smullyan has published two or three books of these logical teasers.
(c) The CH can only hit black squares, so the N must hit b1, and so the CH must deliver the mate. So, 1...Qd4; 2 CHe1, Qf4; 3 CHg7, Qd6; 4 CHa5, Qd4; 5 CHg3, Qb2; 6 Nd2#. The N moving to d2 also allows the CH to mate the Ka1. Neat, eh?
There are magazine and societies devoted to these sorts of things, and a World Solving Championship is hotly contested each year. Two GMs who excel at solving are John Nunn and Jon Mestel.
Problem contacts:
The ProblemistBritish Chess Problem Society
Secretary CAH Russ
76 Albany Drive
Herne Bay
KENT CT6 8JS
Problem Observer
Chess studies
"If a garden is nature perfected, then a study might be described as a game of chess perfected"  .
We have already met a study: Saavedra's position given under 4.4.3.2 [ http://en.wikipedia.org/wiki/Saavedra_position#History ] is a study, that is, a composed position with a realistic feel to it and with reduced material, and there are other studies in this section. Studies are oten referred to a 'endgame studies', and the British study magazine is in fact called EG. As well as the problem magazines above, the BCM publishes a study column each month, and many other fine examples are to be found in Chernev's Practical Chess Endings. Some of the bestknown studies are to be found in The Complete Chess Addict, including the most famous of all  this one of Reti's (1922):

1 Kg7 h4; 2 Kf6 (now 2...h4 allows 3 Ke7, h3; 4 c7, Kb7; 5 Kd7, so..) 2...Kb6; 3 Ke5, Kxc6; 4 Kf4 and draws!
With minimum material, this is a perfect example of a study  it could have arisen in a game, there is no demand to 'win in two moves', and the play has a charming and surprising 'point' to it. One more (Joseph, 1922):

Curios
A related field is not exactly study composition but the challenge of composing to a particular task: for example, from the Usenet newsgroups:From: Fred Galvin <galvin@math.ukans.edu>Newsgroups: sci.math,rec.games.chess.misc
Subject: Re: Strange Question?
Date: Thu, 12 Dec 1996 15:12:07 0600
From: Fred Galvin <galvin@math.ukans.edu>
Organization: University of Kansas Computing Services
On Wed, 11 Dec 1996, Bob Silverman wrote:
> Consider the set of all legally reachable positions in
chess.
> Which position(s) give the largest number of different 1
move
> checkmates? What is that number?
The following positions were the record holders as of January 1, 1969, according to Anthony Dickins, A Guide to Fairy Chess, Second Edition (The Q Press, Richmond, Surrey, 1969); as far as I know, none of them has been *proved* to be best possible; I don't know of any published upper bounds.
Legal position, no promoted men, 47 mates (J. C. West, 1880):
4N1r1/P2PK2P/8/1P1kBQ2/2R5/3N4/B3P3/3R4

Legal position with promoted men, 105 mates (N. Petrovic, 1947):
1B1Q1Q2/2R5/pQ4QN/RB2k3/1Q5Q/N4Q2/K2Q4/6Q1

Illegal position, 143 mates (N. Petrovic, 1947):
QnQQ1QQ1/PQ1Q3Q/QQ4QQ/4k3/QQ4QQ/Q6Q/Q2Q1Q2/1QQQ1QQQ
