A study is a composition in which you do not have to mate in a given number of moves but to win or draw. Studies resemble real endgames, but the solution is always unique and usually surprising.
The solutions have been checked against endgame tablebases — databases that know the exact result of every position with few pieces.
1. The opposition
White to move. Win
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1. Ke5 — White takes the opposition: the kings face each other and the black king has to give way. Any other move draws.
Theme: the opposition.
2. The breakthrough
White to move. Win
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1. b6 — after 1… cxb6 2. a6! bxa6 3. c6 or 1… axb6 2. c6! bxc6 3. a6, one of the white pawns promotes, and the black king is too far away.
3. Réti's study (1921)
One of the most famous studies in history. The white king seems hopelessly far from both its own pawn and Black's.
White to move. Draw
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1. Kg7 — the king walks along the diagonal pursuing two aims at once: to catch the h-pawn and to support its own c-pawn. 1… h4 2. Kf6 Kb6 3. Ke5! h3 4. Kd6 h2 5. c7 h1=Q 6. c8=Q — draw. And on 3… Kxc6 comes 4. Kf4, and the king catches the pawn.
4. The Saavedra position (1895)
White to move. Win
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1. c7 — 1… Rd6+ 2. Kb5 Rd5+ 3. Kb4 Rd4+ 4. Kb3 Rd3+ 5. Kc2 Rd4! 6. c8=R!! Promoting to a queen leads to stalemate after 6… Rc4+ 7. Qxc4. After promoting to a rook White threatens Ra8#, and on 6… Ra4 7. Kb3 wins with a double threat of mate and of taking the rook.