Solving exercises

problem-solving Strategies

When solving problems, one needs a strategy. Next is an adaptation and expansion of Polya's problem-solving strategy from his famous book "How to solve it".

STEP 1. THROW YOUR PEN! AND KEEP SOME DISTANCE

Put down your pen! Resist the urge to start solving right away. Instead, take a moment to pick up your list of exercises and read through them carefully. Don’t try to solve them yet—just approach them with curiosity and focus on understanding each one, especially:

1. Understand every word, concept and property mentioned in the problem. Make sure you can write down the definitions. 

   • While reading the exercise, underline any words you don’t know and look up their definitions in your lecture notes. Avoid Googling them or asking ChatGPT—you’ll only waste time and risk confusion, as definitions and notation may vary.

2. Identify the theoretical concepts required to solve the exercise. Recognize any gaps in your knowledge and review the relevant theory to fill them before attempting the exercise.

STEP 1. GET FAMILIARISED WITH THE PROBLEM 

3. Draw a figure (if you can). 

4. Introduce suitable notation - especially if there are parts of the problem that are expressed in words rather than with mathematical symbology.

5. When dealing with abstract mathematical objects, build concrete examples to familiarize yourself with those objects. Check if what you are trying to prove holds for those particular examples. 

6. For computational-type problems:

   • What is the unknown? What are the data? What is the condition? 

   • Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory? 

   • Separate the various parts of the condition. Can you write them down? 

STEP 2. DEVISE A PLAN / PLAY WITH IT!

1. Could you restate the problem? Could you restate it still differently? Go back to definitions. 

2. Play with it! If you're stuck, take a moment to explore the problem and ask yourself, "What do I wonder?" In other words, what other questions about the objects and properties involved come to mind?

3. If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other? 

4. Think of similar problems that you already know:

   • Have you seen it before? Or have you seen the same problem in a slightly different form? 

   • Do you know a related problem? Do you know a theorem that could be useful? 

   • Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible? 

5. Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem? 

6. For computational-type problems:

   • Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution. 

   • Look at the unknown! Try to think of a familiar problem having the same or a similar unknown. 

STEP 3. CARRY OUT YOUR PLAN

• Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct? 

• Have you used all the assumptions and conditions that appear in the problem? Have you stated clearly where those assumptions have been used in the solution?

STEP 4. LOOKING BACK 

1. Examine the solution obtained. 

   • Is every logical step justified?

   • Is the notation consistent? Did you define the notation?

   • Is your written solution understandable to others?

2. Reflecting on the answer (adapted from Alcock "How to study for a mathematics degree")

   • Why did that procedure work?

   • What could be changed in the question so that it would still work or not work?

   • Can you use the result, the method (or modifications) for some other cases? 

3. Can you check the result? Can you check the argument? 

4. Can you derive the solution differently? Can you see it at a glance? 

STEP 5. STAY ORGANISED

• Ensure your exercises are written clearly and legibly, with each problem statement followed by its solution. Store them in an easily accessible format and organize your work systematically to facilitate the revision process.

solution quality checklist

See the section Basic quality control of WRITTEN solutions - checklist

COMMON ERRORS

Here you can download a document with a list of common mathematical errors. A preview of the first page is visible below:

Further TIPS

• You don’t need to solve the problems in a linear order. If you find yourself stuck on one, move on to the next and return to it later.

• Remember to take breaks. When you start to feel stuck or stressed, pause and take a moment to recover your calm. Only then should you return to work; otherwise, you risk spiraling downward.

Make sure that your final solution is well written (see section 'Writing maths').