| name | math-solving-strategies |
| description | Reference catalog of proof methods, problem-solving tactics, and domain techniques for competition math (induction, contradiction, extremal principle, invariants, etc.). WHEN: only load when strategic guidance is needed, such as pre-planning analysis of a new problem, selecting applicable methods, deadlock mediation, calibration failure diagnosis, choosing a fundamentally different approach after repeated failure. DO NOT USE WHEN: reasoning is executing normally, writing proofs, or running code. |
| user-invocable | false |
Mathematical Problem Solving Strategies
This skill contains general strategies and methods for solving mathematical problems.
Use it for strategic analysis, planning guidance, and method selection.
Proof Methods
Mathematical Induction
Description: Prove base case, assume for n, prove for n+1
When useful:
- Statements involving "for all n ≥ k"
- Recursive sequences or definitions
- Divisibility properties
- Inequalities that grow with n
Variants:
- Simple induction
- Strong induction (assume for all k ≤ n)
- Backwards induction
- Transfinite induction
Proof by Contradiction
Description: Assume the opposite, derive a contradiction
When useful:
- Proving something is impossible
- Proving uniqueness
- Proving irrationality
- "There is no..." statements
Proof by Contrapositive
Description: Instead of P → Q, prove ¬Q → ¬P
When useful:
- When direct proof is awkward
- Statements with "if... then..."
- When negation simplifies the problem
Extremal Principle
Description: Consider minimal/maximal element with the property
When useful:
- Existence proofs
- Optimization problems
- When structure has ordering
Invariants and Monovariants
Description: Quantity that never changes (invariant) or only changes one direction (monovariant)
When useful:
- Process/game problems
- "Can we reach state B from state A?"
- Coloring and tiling problems
Double Counting
Description: Count the same quantity two different ways
When useful:
- Combinatorial identities
- Graph theory (counting edges)
- Proving equalities
Constructive Proof
Description: Explicitly construct the object/algorithm
When useful:
- Existence proofs
- "Find an example..." problems
- Algorithm design
Problem-Solving Tactics
Try Small Cases / Exploratory Analysis
Description: Compute specific cases to find patterns and build intuition
When useful:
- Almost always useful in combinatorics
- Finding the answer before proving
Tips:
- Go far enough to see the pattern. Be aware of small-case anomalies
- When the search space becomes too large, avoid random sampling; instead, observe structural properties from smaller cases to guide exploration (sampling with heuristics, not pure randomness!)
- If problem specifies n with special property (perfect square, prime, etc.), try numbers with similar property. Also try to explore only cases that share relevant properties with the target n.
Special Number Analysis
Description: Competition problems often choose specific numbers because their properties are relevant to the solution. When problem specifies a particular number, analyze its properties
For any number n appearing in the problem, check:
- Is it a perfect square?
- Is it a perfect cube or higher power?
- Is it prime or has special prime factorization?
- Is it Powers of 2?
- Can it be expressed as a sum? (e.g., 2016 = 1+2+...+63)
Work Backwards
Description: Start from the goal and work towards given conditions
When useful:
- Construction problems
- When end state is well-defined
- "Reach configuration X" problems
Symmetry
Description: Exploit symmetry to simplify or count
When useful:
- Geometric problems
- Counting problems with symmetric objects
- Function equations
Greedy Algorithm Analysis
Description: Consider what greedy approach gives, then adjust
When useful:
- Optimization problems
- Game theory
- Resource allocation
Case Analysis
Description: Split into exhaustive cases
When useful:
- When problem has natural divisions
- Parity arguments (odd/even)
- Modular arithmetic
Generalize then Specialize
Description: Prove a more general statement, then apply to specific case
When useful:
- When specific case is awkward
- Induction-friendly reformulation
Add Auxiliary Elements
Description: Add points, lines, variables to reveal structure
When useful:
- Geometry problems
- Algebraic manipulation
- Creating useful equations
Construction Search
Description: When asked for a sharp / largest / smallest / optimal value, the correct answer is determined by the most extreme valid construction. Searching only one family (e.g. "the obvious linear / chain / repeating one / identity") can lock in a wrong target. Before committing to a value, try construction families orthogonal to the first one you found.
Generic axes to vary:
- Dimension / density: 1-D chain → 2-D filling → 3-D / layered. Denser packings often shift the extremal ratio sharply. For symmetric properties, also consider from 1D (left/right) to 2D (North/East/South/West) and even higher dimensions.
- Recursive / fractal / self-similar: iterated substitution rules; their limits frequently exceed any constant from a fixed family.
- Asymmetric / symmetry-broken: non-symmetric extremals can beat symmetric ones in combinatorial problems.
- Witness set: construct a set that must satisfy some property. Try elements adjacent to special objects.
- Boundary / degenerate parameters: drive a free parameter to 0, infinity, or a critical threshold; "interior" intuition can fail there.
- Concatenation / gluing: combining two valid objects can produce a ratio neither piece achieves alone.
- Random / probabilistic: averaging over random samples may reveal that the typical case differs from the obvious extremal candidate.
When useful:
- Sharp inequality / tight bound problems
- "Largest k such that ..." / "smallest n such that ..."
- Whenever your first candidate construction is "obvious" — that is exactly when a different family is most likely to beat it
- After repeated proof failure on a sharp claim — strongly consider the claim itself is wrong before patching the proof again
Discipline: A construction is only an extremal candidate until it has been stress-tested against at least one orthogonal family. Tag a one-sided value as [Conjecture], never [Lemma], until both directions are proved.
Number Theory
Modular Arithmetic
Description: Work modulo small primes or relevant moduli
Tips:
- Try mod 2 (parity), mod 3, mod 4, mod small primes
- Chinese Remainder Theorem for composite moduli
- Fermat's Little Theorem for prime moduli
Prime Factorization
Description: Factor numbers and work with exponents
Tips:
- Unique factorization is powerful
- Count prime factors (with/without multiplicity)
- Legendre's formula for factorials
Divisibility Analysis
Description: Study what divides what
Tips:
- gcd and lcm properties
- Euclidean algorithm
- Bézout's identity
Diophantine Equations
Description: Integer solutions to polynomial equations
Tips:
- Factorization methods
- Descent arguments
- Modular constraints
Combinatorics
Bijection
Description: Establish 1-1 correspondence with easier-to-count set
When useful:
- Counting problems
- Proving two quantities equal
Generating Functions
Description: Encode sequence as power series
When useful:
- Recurrence relations
- Counting with constraints
- Partition problems
Inclusion-Exclusion
Description: |A ∪ B| = |A| + |B| - |A ∩ B|, generalized
When useful:
- Counting with "or" conditions
- Derangements, surjections
Recursion
Description: Express answer for n in terms of smaller cases
When useful:
- Almost all counting problems
- Dynamic programming
Graph Theory Translation
Description: Model problem as a graph
When useful:
- Relationship problems
- Matching, coloring
- Connectivity questions
Pigeonhole Principle
Description: n+1 objects in n boxes → some box has ≥2
When useful:
- "At least one..." statements
- Existence proofs
- Coloring/assignment problems
- Infinite sequences with finite states
Variants:
- Simple pigeonhole
- Generalized (kn+1 objects, n boxes → some has ≥k+1)
- Infinite pigeonhole
Algebra
Polynomial Analysis
Description: Roots, coefficients, factorization
Tips:
- Vieta's formulas
- Symmetric polynomials
- Rational root theorem
Inequalities
Description: AM-GM, Cauchy-Schwarz, Jensen, etc.
Tips:
- AM-GM for products/sums
- Cauchy-Schwarz for inner products
- Jensen for convex/concave functions
- Rearrangement inequality
Functional Equations
Description: Find all functions satisfying given condition
Tips:
- Try special values (0, 1, -x)
- Injectivity/surjectivity
- Continuity assumptions
Common Pitfalls
| Pitfall | Context |
|---|
| Off-by-One Errors | Counting, indexing, boundaries |
| Missing Edge Cases | n=0, n=1, empty set, single element |
| Assuming More Than Given | Integers vs reals, positive vs non-negative |
| Circular Reasoning | Using what you're trying to prove |
| Incomplete Case Analysis | Forgetting cases in exhaustive proofs |
| Ignoring Constraints | Bounds, divisibility requirements |
Conjecturing Discipline
Avoid Premature Pattern Extrapolation
Description: Multiple formulas can fit a small number of cases equally well, then diverge dramatically at larger values.
When to be careful:
- The conjecture was derived from fewer than 8-10 data points
- The problem involves a number with special structure
- Different candidate formulas (e.g., linear vs sublinear) both fit small cases
Tips:
- Always test at least two plausible candidate formulas
- Use constructions (not just exhaustive search) to verify larger cases
- If small-case formulas seem too simple, seek structural insight from the problem
- Exploit special structure in problem parameters (factorizations, divisibility, geometric properties)
- Verify both construction AND lower bound before committing to a conjecture
