How to solve a problem you have not seen before

Restate the problem in your own words. Write down what is given, what is asked for, and what constraints apply. If you cannot state the goal in a sentence, solving is premature.

Draw something. A diagram, a table, a number line. Unfamiliar problems are frequently familiar problems in unfamiliar clothing, and a drawing is what strips the clothing off.

• Try a smaller case. If the question is about a hundred terms, work out three and look for the pattern.
• Work backwards from the answer's required form.
• Look for an invariant, something that stays fixed while everything else changes.
• Consider extreme or trivial cases to bound what the answer can be.
• Introduce a variable for the thing you want and write down every relationship it satisfies.
• Ask what related problem you already know how to solve, then adapt it.

Being stuck is information, not failure. Identify what specifically is blocking you: a missing fact, an unclear goal, or an algebraic dead end. Each calls for a different response, and naming it converts frustration into a next step.

If nothing moves, change representation. Turn an algebra problem into a graph, a geometry problem into coordinates, a word problem into a table. A problem that resists one representation often collapses in another.

After solving, check the result and then ask two further questions: is there a shorter route, and what class of problem did this turn out to be? That second question is what converts a solved exercise into transferable knowledge.

Keep a short record of problem types and the move that unlocked each. Over a term this becomes a personal index that is far more useful than a pile of worked solutions.

Rereading a worked solution feels productive because it feels easy, and that feeling is the problem. Recognition is not the same as the ability to produce. Cover the solution and attempt the problem cold, then compare.

Mix topics when practising rather than doing thirty problems of one type. Interleaving is harder and scores worse during practice, but it produces markedly better performance later, because exams do not tell you which method applies.