吳清源
摘自《天才的棋譜》
下面是吳清源談自己的思考方法，己多次出現在各種棋譜，請看：
「我們這些專業棋手在對方落子時，腦子裏立即閃出第一感，一、二、三、四、五......各種應法都自然涌現出來。以我來說，首先選出感覺最好的一步，然後加以考慮，一旦覺得這一步可以，再剩下的應法就不再考慮了，所以下得快。但木谷【實】先生卻截然相反。他說他是從感覺最不好一箸開始考慮，這樣就不會算漏。」
他也承認不考慮其他應法會錯過更好的棋，但他說出為甚麼，請看：
「......我這種人考慮過多，反而容易迷惑。所以盡管我經常錯過好手，也不敢考慮太多。」
- 摘自《勁敵木谷實》一章
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M.M. Botvinnik
譯自《Selected Games 1967-1970》
六十年代的國際象棋世界冠軍 Botvinnik ，也是研究電腦奕棋的先鋒。他寫出人下棋時的思考過程和電腦的思路比較，請看：
「...讓我們看棋手的思考過程：棋手不能算盡所有變化，也不會考慮到所有可走的箸法...」
「...思考棋局時，棋手大致會分析兩到四個走法，憑直覺，按經驗挑選。如果平均一局棋(指國際象棋)四十回合(白一步加黑一步)，棋手每局會分析大約一百步棋...」
「當然，每局棋棋手不只考慮一百箸，他分析的更多。如果平均來說，棋手算2~3回合深的話，分析過的箸法數目就會很大。」
「棋手計算時不會注意到棋盤上全部64格。這樣計算就會簡單些。例如每次在腦海考慮到8~16格，推算起來就會輕鬆幾倍。」
「也要提出的是，棋手不會注意到棋盤上所有棋子。全部25~30個棋子當中，棋手只注意3~6個，這樣，推算工作就會進一步簡化。」
「因此，每局棋棋手只會考慮到有限的棋子、有限的格，只會分析那些直接衝突的棋子，和有可能發生衝突的格...」
「但是，棋手怎麼知道他選的棋子和格子是正確的呢？有個方法，我(Botvinnik)認為所有大師都會用，就叫『反覆檢查近似法』吧。大師選一箸，加以推算。如果推算時發現新的棋子和格子，就會用於下次推算，如此循環。推算幾次後，他就有足夠﹝也許不足﹞的準確度去決定那些相互作用的棋子和格子。大師就是這樣通盤推算的。」
「大師推算5~6回合深。」
- 譯自《The Tale of a Small Tree》
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Kotov
摘自《Think Like a Grandmaster》
這本書是國際象棋經典著作，Kotov 認為要棋下的好，就要 discipline 自我思考方式。他的書第一、二章提出棋路樹﹝Tree
of Analysis﹞的思考方法，撮要如下：
〈一〉心裏列出侯選箸法﹝candidate moves﹞；
〈二〉依次推算，先算好一路變化再算另一路；
〈三〉每一路變化只推算一次；
〈四〉選出最佳棋步﹝正式比賽時在記錄紙上寫下棋步﹞；
〈五〉重新看看有沒有大漏著；
〈六〉走子 。
Kotov的棋路樹思考法，今天的棋手，例如特級大師 John Nunn ，提出不少補充。
這本經典著作，比較少人注意到 Kotov 論高手怎樣思考實戰殘局。一次大師聚會拆棋，拆不通某殘局，世界冠軍
Capablanca剛巧經過，大家向他請教。Kotov注意到世界冠軍推敲殘棋，是用「重新分佈棋子」法，請看：
「......Once in a lobby of the Hall of Columns of the Trade Union Centre
in Moscow a group of masters were analysing an ending. They could not
find the right way to go about things and there was a lot of arguing about
it. Suddenly Capablanca came into the room. He was always fond of walking
about when it was his opponent's turn to move. Learning the reason for
the dispute the Cuban bent down to look at the position, said, 'Si, si,'
and suddenly redistributed the pieces all over the board to show what
the correct formation to show what the correct formation was for the side
that was trying to win. I haven't exaggerated at all. Don Jose literally
pushed the pieces round the board without making moves. He just put them
in fresh positions where he thought they were needed.
Suddenly everything became clear to us. The correct scheme of things
had been set up and now the win by Capablaca's mastery, and soon had further
evidence of the need to think schematically about the ending.」
後來 Kotov 翻閱 Capablanca 的棋譜，發現 Capablanca 下殘棋時常用「棋子分佈法」思考，並舉了
Capablanca 一盤棋為例，在棋評中 Kotov 寫到：
「......In [a] position Capablanca again drew up a winning scheme, namely
Queen on e3, Rook on c3, knight on d4 and pawn on b4. Separate moves did
not interest him as much as the necessary scheme, and in the subsequent
play everything went as the had planned.」
以後 Kotov 發現不少殘棋高手都用同「棋子分佈法」思考：
「......Ever since then I gave watched carefully how the great endgame
players handle, or rather, regard the ending. I gained a great deal from
my friendship with Sergei Belavenets......I saw how Belavenets always
thoughts about the ending in term of schemes, in terms of the layout of
one's forces. Later I noticed the same type of thinking in the great modern
endgame masters Flohr and Smyslov.」
- 摘自《The Ending》一章
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John Nunn
摘自《Secrets of Practical Chess》
英國特級大師 John Nunn 多次代表英國參加 Chess Olympaid，奪過四面金牌。他的箸作條理分明，解說清楚。是西方極有名的棋書作者。這本書第一章談他對算度的看法。下面摘錄幾段供大家欣賞。
John Nunn 從 Kotov 的著作談起，Nunn
肯定 Kotov 提出的棋路樹思考法是重要的思考法，並指出：電腦﹝計算機﹞單憑棋路樹思考法，就有很強的棋力。﹝按：Nunn 指的是國際象棋軟件，今天，在一般
PC 上運行的，也很接近特級大師水平。﹞
「...the 'Tree of Analysis' is a very important method of chess
thinking; computers have shown that it is possible to play very strongly
using practically nothing else. Kotov's book described in detail the mental
processes involved in concrete analysis. If we assume that White is to
move, then Kotov recommended that White form a list of 'candidate moves'
which he is trying to decide between, making sure that the list is complete.
For each of these, White creates a list of possible replies by Black and
so on, following each branch in turn until a definite evaluation can be
given...」
然後，Nunn 指出棋路樹的限制：人類不可能單純運用棋路樹計算，每步棋都有幾十種可能，棋路太多，人類負擔不了：
「...there are, typically, dozens of legal moves in an average chess
position, a tree including very legal move rapidly becomes too dense for
human beings to handle.」
當然，不必每種可能都去想，想合理的棋就行了。可是，就是每步只有五個合理選擇，三個回合下來，仍然有萬餘條棋路：
「...Of course, it isn't necessary to consider every legal move, since
a considerable percentage of these legal moves are nonsensical, and this
tends to thin the tree somewhat. Nevertheless, even if there are only
five reasonable possibilities at each ply (we will borrow a term from
the computer chess world and call half a move a ply), after three whole
moves ﹝一 move 指黑、白各走一步，即一回合﹞there are 15,625
'leaves'...」
單純的棋路樹只適用在合理選擇相對少的局面，實際上，就是戰術局面和某些殘局：
「 ...It follows that analysising
solely by means of an analytical tree is only possible when the number
of reasonable possibilities for the two players is limited - in practice
this means tactical positions and certain types of endgames ... 」
其餘局面， 除運用棋路樹外，還要考慮其他因素：
「... one constructs some sort of analytical tree in thinking about almost
any position; in less tactical situations, where the opponent's replies
are much less predictable, one would not rely solely on the 'tree' but
also take other factors ﹝例如審局﹞into consideration.」
Nunn 特別請讀者注意 Kotov 的遂條棋路計算理論：算一路棋時，不要跳去想別一路棋的警告：
「 Kotov specifically warned against jumping from branch to branch; he
thought that you should analyse each candidate move until it can be definitely
evaluated, and only then move to the next one.」
Nunn 也指出遂條棋路計算的壞處：沒有效率。假如對手有甲、乙兩路應法。運用 Kotov
棋路樹，先算甲棋路，二十分鐘後覺得己方稍優；然後才計算乙棋路，幾分鐘後發現對手佔優。二十分鐘就浪費掉了。
「The most obvious is that it can be extremely inefficient. Let
us suppose that you are analysising a possible combination 1 Bxh7+ Kxh7
2 Ng5+. There are two defences, 2...Kg6 and 2 Kg8. You start by analysising
2 Kg6; it is very complicated, but after twenty minutes you decide that
White has the advantage. Then you start looking at 2 Kg8. After a couple
of minutes it becomes obvious that this refutes the sacifices. The upshot
of Kotov's inflexible approach is that you have lost twenty minutes' thinking
time, solely on account of the bad luck of having chosen the wrong move
to analyse first....」
比較合理的計算方法，是先用幾分鐘分別看兩種應法。可能很快就看出己方的設想不行，然後可以找別棋：
「...A more rational approach would be to spend a couple of minutes looking
at each of the two alternatives. It is possible that this will reveal
one of them to refute the sacrifice, at which point Kxh7+ can be abandoned.
It is also possible that the preliminary analysis will show one of them
to be a clear-cut loss, in which case you can switch attention to the
other one in the confidence that it is definitely the critical line. 」
Nunn 還指出，各棋路不是獨立的。推算第二路棋時有新想法時，回頭想第一路棋也是合理的：
「...(Kotov's method) fails to take into account
the synergistic effect of analysing several lines. The analysis of move
A and move B are very often not independent of one another. Suppose you
have rejected move A, but when analysing mvoe B you suddenly notice a
tactical possibility. It makes sense to return to move A to see if the
same possibility is applicable here.」
Nunn 的結綸是： Kotov 棋路樹不可呆板運用。必須用其他思考法補充。尤須注意運用「快速溜灠法」：
「...I favour the 'quick scan' approach, looking briefly at all the major
lines to see if any can be quickly be resolved. With any luck, this will
be enough to settle the analysis; if not, at least you know where the
tricky areas lie.」
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Adrian de Groot
﹝不少心理學書都有提及，並非來自某一本﹞
de Groot 既是大師﹝1937年曾代表荷蘭出戰 Olympiad﹞，又是專業心理學家。他是研究「棋手怎樣思考」的先鋒。後來的心理學者都以他的實驗為典範。
他擺出棋局(按：他本人的比賽時遇到的局面)，要求參加實驗者找出最好續箸，並要開口思考，即邊思考邊說出想法(按：請人速記下參加者的說話)。參加實驗者包括當時的特級大師，Keres，Alekhine，Flohr，Fine，Euwe
，一些大師和專家棋手。他發現特級大師、大師和專家棋手計算的深度相差無幾，思考過程也沒大分別。主要的分別是：高手會計算較好棋，低手花時間去計算劣箸。因此，他認為計算能力不是高低手的主要分別。
他的另一個實驗，是擺出棋局給實驗者看2~10秒，然後要實驗者在憑記憶在空棋盤上重擺。他發現棋藝越高，重擺的棋局越準確。特級大師和大師的成績是91.4%，專家棋手69.6%，再弱一點的棋手只有52.5%。
他的結論是：「The Master does not calculate more than the
Expert...: the Master sees more than the Expert, especially the
more important things」
要注意的是：他的結論是錯的。另有科學家﹝例如Holding and Reynold﹞實驗証明 高手確實比弱手推算得深。
de Groof擺的棋局，可以在http://www.ex.ac.uk/~dregis/DR/webb2.html找到。
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Chase and Simon
摘自《Chess Psychology》
Chase and Simon 是 Carnegie-Mellon University 的學者，他們希望找出為何在 de Groot 的記憶實驗中，大師看棋局幾秒，就能記住？是不是大師的記憶力特別好？
他們的實驗，請象棋大師、專家棋士、初學者看了實戰棋局幾秒後憑記憶重擺棋局。發現如果選用實戰棋局試驗，棋力越高的，重擺得越準；但用電腦隨機擺出的棋局﹝用實戰棋局相同的棋子﹞，三組都同樣的差，比初學者重擺實戰棋局更差。
Chase and Simon 認為，棋手的成就是來自長期記憶裏的大量相似棋形，棋手並非從頭開始記憶棋局，而是下意識和腦子裏的相似棋形比較。
Chase and Simon 另外做過一個實驗，他們擺出實戰棋局，要求受實驗者在用盡快用另外一副棋子在旁重擺棋局。發現大師每次回頭看原棋局，只看兩秒；初學者每次看四秒。他們認為這是大師腦裏有較多「棋形結構」之故。
腦中棋形當然是長期研究棋藝而來，兩位學者用科學方法證明了學習和經驗對棋藝的重要，這個實驗結果也可以解釋為甚麼沒有天生的大師，也可解釋為何天才棋手也要學棋多年才可以成為大師。
另請看 Holding and Reynolds 的實驗，證明腦中棋形並非高低手唯一分別。
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V. Malkin
摘自《Chess Psychology》
Malkin 是蘇聯心理學者，他也做個實驗研究棋手的思維方式。
他擺出只有一個答案的戰術組合棋局，並且在棋局裏加入了一枚和戰術組合無關的棋子。要求受實驗者找出答案，然後憑記憶重擺棋局。受實驗者有特級大師﹝Tal
和 Vasyukov﹞、大師、候選大師。
兩位特級大師很快就找出答案，並準確重擺棋局。其他人要用稍長的時間才找到答案，重擺棋局時記錯那枚額外棋子的位置。
Malkin 認為那額外的棋子，由於沒有和棋手腦中的棋形吻合。棋手重擺時就把它放在「應該」存在的格子裏，這就是大漏和盲棋的來源。
Malkin 的實驗支持 Chase and Simon 的結論。
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Holding and Reynolds
摘自《Chess Psychology》
Holding and Reynold 把 Chase and Simon 的實驗更進一步，Holding and Revnold 用電腦擺出隨機棋局﹝棋子都在合法位置﹞，要求受實驗者憑記憶重擺棋局，然後為棋局找出最好續箸。參加實驗的有國際象棋大師和美國有等級分的棋手。
重擺棋局的結果和 Chase and Simon 的一樣；走子分析部份高低手大有分別，棋越高分析得越好。
Holding and Reynold 認為以往的研究對於「高低手的分別在於腦中棋形的多少」這結論有局限，因為重擺棋局的結果證明大師對於隨機棋局，腦子裏沒有相似棋形，如棋力只和腦中棋形有關，大師和弱手的分析應該是一樣差的，但實驗結果卻是大師分析得較好，證明棋力確和計算能力有關。
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修訂：
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