When you’re cruising through a set of problems and suddenly hit a wall, it can feel like you’ve run into an impenetrable barrier. But don’t worry—getting stuck is part of the problem-solving process. The key is knowing how to navigate through it.

This guide provides a structured approach to breaking through mental blocks and getting back on track.

0. Find What the Question Is Asking For

Strip the problem to its core.

Before rushing into calculations, take a step back and clarify the problem’s objective:

  • Rewrite the goal in clear mathematical notation. If the problem says, “Find all real solutions,” explicitly state:
    $$ \text{Solve for } x \in \mathbb{R}: \quad [\text{equation}] $$
  • Identify key verbs such as “Prove,” “Compute,” “Determine,” or “Describe,” as each suggests a different approach.
  • Restate the problem in your own words. If you struggle to do this, you likely don’t fully understand it yet.
  • Confirm definitions. Ensure you’re interpreting every term correctly, such as “distinct integers” or “cyclic quadrilateral.”

Clarifying the problem is often half the battle.

1. Read the Problem Again, Carefully

Details matter.

  • Highlight constraints: Watch for keywords like “positive integers,” “at most,” or “distinct.”
  • Distinguish quantifiers: “For all $ x $” means something very different from “There exists an $ x $.”
  • Rewrite conditions symbolically. For example, if a problem states, “The product of two numbers is 12,” write:
    $$ ab = 12 $$ This is pretty much the singular most generally useful trick, as writing thigns algebraically will allow you to recall facts and apply them to the problem at hand.
  • Be wary of tricky phrasing. Words like “at least,” “exactly,” and “in terms of” can change the nature of the problem.

A careful rereading often reveals overlooked insights.

2. Visualize the Problem

Turn abstractions into concrete models.

  • Draw diagrams for geometry and combinatorics problems. If working with graphs, sketch nodes and edges. Give yourself some concrete examples to work with.
  • Plot functions or construct tables to organize data in algebra or sequences.
  • Use number lines or modular grids for number theory problems (e.g., visualizing modular arithmetic).
  • Create step-by-step flowcharts for process-based problems (e.g., algorithmic puzzles).

Many problems become trivial once represented visually.

3. Modify the Constraints

Experiment with flexibility.

If you’re stuck, play with the problem’s constraints to understand its structure better.

a) Simplify

  • Set variables to small values like $0, 1,$ or $2$.
  • Solve a special case (e.g., if $ n $ is general, try $ n = 2, 3 $).

b) Generalize

  • Replace numbers with variables to explore broader patterns.
  • If the problem involves integers, what happens with rationals or reals?

c) Consider Extremal Cases

  • What happens when $ x = 0 $ or $ x \to \infty $?
  • Assume equality in inequalities—often a key insight.

d) Swap Conditions

  • In geometry, try replacing “circle” with “sphere” to test if the property still holds.

4. Experiment and Iterate

Learn by doing.

  • Guess and Refine:
    • Assume symmetry in variables and check for contradictions.
    • Use approximations: $ \sqrt{20} \approx 4.5 $, $ \pi^2 \approx 10 $.
  • Iterate Small Cases:
    • Compute the first few terms of a recurrence relation manually.
    • Try step-by-step adjustments, similar to gradient descent in optimization.
  • Track Patterns:
    • Identify repeating remainders in modular arithmetic.
    • Look for cycles in sequences or function iterations.

5. Work Backwards

Start from the conclusion.

  • Reverse-engineer the equation. If proving $ \sin^2 x + \cos^2 x = 1 $, try deriving it from definitions.
  • Use the desired conclusion to shape your approach. What must be true for the statement to hold?
  • Construct backwards. If you’re asked to find a sequence, start with the answer and determine how it could be generated.

6. Exploit Symmetry and Invariants

Find what doesn’t change.

  • Use symmetry in variables: Assume $ a = b $ in symmetric equations to simplify.
  • Look for invariants:
    • Parity (odd/even), modular residues, or conserved quantities.
    • Example: In a game with alternating moves, parity often determines the winner.
  • Use duality: Swap reciprocals ($ x \leftrightarrow \frac{1}{x} $) or apply transformations in geometry.

7. Leverage Known Theorems and Formulas

Stand on the shoulders of giants.

  • Common techniques:
    • Geometry: Power of a point, Ceva’s theorem.
    • Algebra: Vieta’s formulas, AM-GM inequality.
    • Number Theory: Bezout’s identity, Chinese Remainder Theorem.
  • Check applicability:
    • Do the problem’s conditions meet the theorem’s requirements?
    • Example: To use the Pigeonhole Principle, identify “pigeons” and “boxes.”

8. Divide and Conquer

Break the problem into manageable pieces.

  • Split the problem into subcases.
    • Handle even and odd $ n $ separately.
    • Convert a complex sum into multiple smaller sums.
  • Hybrid Approaches:
    • Convert geometry problems into algebraic coordinate-based problems.

9. Re-examine Your Approach

If you’re stuck, pivot.

  • Try a different proof method. Switch from induction to contradiction.
  • Question your assumptions. Are you assuming something that isn’t given (e.g., integers vs. reals)?
  • Take a break. Returning with fresh eyes often reveals missed insights.

10. Verify and Validate

Ensure your solution holds up.

  • Check dimensions (if applicable). Do units match?
  • Test edge cases. What happens for $ n = 0 $ or when two variables are equal?
  • Substitute back into the original problem. Does it satisfy all constraints?
  • Perform sanity checks.
    • If you found a probability greater than 1, something went wrong.
    • If you found a triangle with sides $ 2, 2, 5 $, recheck your inequalities.