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I know that research math problems are unsolved and unseen, while contest math problems can be solved. All bolds are mine.

  1. How do 'tricks' in research math differ from those in contest math?

  2. Do the meanings of 'tricks' differ between the 3 citations? The 1st two argue against 'tricks' (in contest math) but Prof. Gowers refers to 'tricks' (in research math)?

In The Map of My Life [...] Goro Shimura wrote of his experience teaching at a cram school

I discovered that many of the exam problems were artificial and required some clever tricks. I avoided such types, and chose more straightforward problems, which one could solve with standard techniques and basic knowledge. There is a competition called the Mathematical Olympic, in which a competitor is asked to solve some problems, which are difficult and of the type I avoided. Though such a competition may have its raison d'être, I think those younger people who are seriously interested in mathematics will lose nothing by ignoring it.

[...]

In his Mathematical Education essay, Fields Medalist William Thurston said

Related to precociousness is the popular tendency to think of mathematics as a race or as an athletic competition. There are widespread high school math leagues: teams from regional high schools meet periodically and are given several problems, with an hour or so to solve them.

There are also state, national and international competitions. These competitions are fun, interesting, and educationally effective for the people who are successful in them. But they also have a downside. The competitions reinforce the notion that either you ‘have good math genes’, or you do not. They put an emphasis on being quick, at the expense of being deep and thoughtful. They emphasize questions which are puzzles with some hidden trick, rather than more realistic problems where a systematic and persistent approach is important. This discourages many people who are not as quick or as practiced, but might be good at working through problems when they have the time to think through them. Some of the best performers on the contests do become good mathematicians, but there are also many top mathematicians who were not so good on contest math.

[...]

In his book Mathematics: A Very Short Introduction, Fields Medalist Timothy Gowers writes

This last quality is, ultimately, more important than freakish mental speed: the most profound contributions to mathematics are often made by tortoises rather than hares. As mathematicians develop, they learn various tricks of the trade, partly from the work of other mathematicians and partly as a result of many hours spent thinking about mathematics. What determines whether they can use their expertise to solve notorious problems is, in large measure, a matter of careful planning: attempting problems that are likely to be fruitful, knowing when to give up a line of thought (a difficult judgement to make), being able to sketch broad outlines of arguments before, just occasionally, managing to fill in the details. This demands a level of maturity which is by no means incompatible with genius but which does not always accompany it.

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  • $\begingroup$ “Tricks of the trade” means techniques that are known to all specialists. That’s quite different from the use of the (unmodified) word “trick” in the first two quotations. $\endgroup$ – Lubin Dec 19 '17 at 3:14

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