...José Raul Capablanca how many moves ahead he looked while playing, Capablanca replied: "Only one, but it's always the right one." And he was, for the most part, telling the truth.
The most famous mate-in-X-number-of moves announcement was made by a woman player in the early 20th century. I am sorry to say I cannot recollect her name, but I think she correctly predicted forced mate over 20 moves in advance. Thinking ahead ten moves with clarity would, of course, be impossible, since the dauntingly complicated move-trees change dynamically with each potential candidate response at each successive ply. Such achievements are possible only when the sequence envisioned is comprised of forcing moves such as check. Without a narrow range of forcing moves, millions of possibilities can't be effectively reduced to a viable sequence leading to a definite result.
The most effective players employ refined pattern-recognition skills to evaluate the position and intuit candidate moves. Then analog "if I move here and he moves there" calculations are attempted in an effort to corroborate the intuited candidates. With rare exception, the original intuited candidate move proves to be the soundest choice.
Some interesting chess quotes: http://www.sonic.net/finearts/tempimgs/chesquot.htm
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