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Probably most people will acknowledge the importance of doing exercises when reading a mathematical textbook. Here I am talking about a textbook of similar level as those ones listed in GTM. However, there is usually no solution avalible for those exercises in most mathematical books. When one tries to do exercises without solutions, it seems to me that two problems may occur:

  1. One cannot easily verify if his/her answer is correct. For example, if an exercise asks the reader to find all groups of order $52$, without a solution how can one make sure that s/he has listed all the possible answers without mistake? Even for those "show that" questions, it is still possible that the reader gives a false proof and contains some (possibly non-obvious) errors.

  2. For those questions one just cannot solve after working on it for a long time, how can s/he find out the solution? I know that one can ask the question here or to a teacher or a fellow student, etc.; but is there any better way?

For myself, I find much more motivated to do those exercises with solutions, then compare my answers with the suggested ones, then continue to do more. But for those without solutions I can hardly even motivate myself to attempt them, for the two reasons mentioned above. Anyone has any suggestions on what we should do with those exercises in a textbook?

Any comments and suggestions will be much appreciated.

Zuriel
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2 Answers2

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I think that it would be nice to have a separate book with full workout solutions. Remember that sometime there is more than one choice to get to the right answer.However the person that is making use of this book will have the option to choose the best way he understands the most and that he or she can easily remember. I think that every one should be mature enough not to cheat on himself,and use such a book as a tool to check his abilities.

Carmel Vella
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    Thanks! However, is such a book always available for all subjects? I know that for more elementary subjects such as calculus or linear algebra this should not be an issue. But how about for algebraic topology or knot theory? – Zuriel Mar 01 '18 at 08:23
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For exercises with solutions, there is a terrible bias : you work much less to find the solution.

The thing is that you don't really need to find the solution of an exercise, a good exercise should let you think a lot of the profound nature of the problem.

The exercises without solutions are the best to learn, by far.

And after some days, if you still don't find the solution and you are frustrated, ask a fellow. It encourages discussion which is really good for learning.

Your problem $\#1$ is a fake problem, because at the end when you write a proof, you know if there is something that is not perfectly clear in your proof. Once you've learnt how to write a perfect detailed proof, there is no space for mistakes.

Damien L
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    I agree completely. As an additional solution for #2, note that you can always post a question on SE. I am aware also of compilation of answers for some textbooks that do not normally have answers (for instance, Hartshorne's Algebraic Geometry). – Jakub Konieczny Mar 16 '13 at 13:12
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    I think this answer contains several oversimplifications. First: "The thing is that you don't really need to find the solution of an exercise". But sometimes you really do: many texts leave steps of theorems as exercises. If you can't do the exercise, what do you do, stop reading the text? "Once you've learnt how to write a perfect detailed proof, there is no space for mistakes." It sounds to me like you are saying that sufficiently mature mathematicians never write faulty proofs. This is a nice idea but it is unfortunately false: have you ever refereed a research paper? Or written one? – Pete L. Clark Jul 02 '13 at 11:08
  • @JakubKonieczny, may I know where I can find the "compilation of answers for some textbooks that do not normally have answers"? – Zuriel Sep 20 '16 at 01:41
  • Thanks for the answer! May I know if it is normal for the author(s) of the book have all the solutions and will be willing to share if we ask for it? – Zuriel Dec 27 '16 at 23:54
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    @JakubKonieczny Without solutions I make too less exercises and get in time stress, because I get stuck on one for too long and can't ask too much to teachers. In addition, on SE most of my questions have not been answered. For projects it is normal that not every problem can be solved, but there is a difference in math subjects, where learning maths by making much exercises is of highest priority, and projects, where doing experience with research has priority. – Rocco van Vreumingen Apr 06 '19 at 19:32