claude-plugins-official/plugins/math-olympiad/skills/math-olympiad/references/solver_heuristics.md

Solver Heuristics (Pólya + Olympiad Practice)

For solver subagents. These are the moves to try when the direct approach stalls.

Pólya's core moves (from "How to Solve It")

Have you seen a related problem? Not the same problem — one with the same UNKNOWN, or the same STRUCTURE. A problem about covering points with lines has the same shape as one about covering lattice points with arithmetic progressions.

Specialize. If you can't solve the given problem, solve n=3, n=4, n=5 by hand. The pattern is often the proof. (But: test past the first nontrivial case — n≤3 may be degenerate.)

Generalize (inventor's paradox). The more ambitious problem sometimes has MORE structure and is easier. "Prove for all primes" might be harder than "prove for all integers" if the integer case has a clean induction.

Drop a condition. What happens if you relax one hypothesis? Does the result become trivially false? Where? That WHERE is often the key step — the point where the condition is load-bearing.

Work backwards. Start from what you want to prove. What would imply it? What would imply THAT? If this chain meets something you can prove directly, you have the proof (reversed).

Auxiliary element. Introduce something not in the problem — a new variable, a reflection, a well-chosen function. Olympiad geometry lives on this (auxiliary points, circles).

Olympiad-specific moves

Find the invariant. If there's a process (game, transformation, iteration), what quantity is preserved? Parity, sum, product modulo something.

Find the extremal. Take the LARGEST, or SMALLEST, or LEFTMOST object. Extremal choices often have extra properties that generic choices don't.

Double count. Count the same thing two ways. Incidences, edges, sums over pairs.

Coloring / parity. Can you 2-color the objects so the claim becomes a parity statement?

Smoothing / adjusting. For inequalities: if you perturb two variables closer together (or further apart), does the expression increase or decrease? Extremize.

Symmetry → WLOG. If the problem is symmetric in x,y,z, you can assume x≤y≤z. But only if the conclusion is ALSO symmetric.

Geometry-specific moves

Standard angles (induction, invariants, extremal) are often wrong-shaped for olympiad geometry. Use these instead:

Coordinate bash. Place the configuration in coordinates. Choose them to kill degrees of freedom (origin at a center, axis along a line). Grind out the algebra. Ugly but reliable.

Auxiliary point. Introduce a point not in the problem — a reflection, a second intersection, the point where two lines "should" meet. Often the key construction is finding the right extra point.

Power of a point. For any point P and circle ω, PA·PB is the same for every line through P meeting ω at A, B. Use it to turn ratios into equalities.

Spiral similarity / rotation. Two directly similar triangles are related by a spiral similarity (rotation + scaling about a fixed point). Find that point — it often lies on a circle you already have.

Inversion. When there are many circles or tangencies, invert about a well-chosen center. Circles through the center become lines; tangencies become simpler tangencies.

Angle chase. Cyclic quadrilaterals give equal angles. Tangent-chord gives an angle equal to the inscribed angle. Chase around the figure.

Geometry-specific moves (these are DIFFERENT)

The standard angles (invariant, extremal, induction) don't fit circles/circumcenters/orthocenters. Geometry needs:

Coordinate bash. Place one point at origin, another on the x-axis. Compute everything explicitly. The algebra is heavy but mechanical. For two circles with centers M, N and radii r, R: set M=(0,0), N=(d,0), then the intersection points have x-coordinate (r²+d²−R²)/2d and everything follows.

Auxiliary point. Introduce a point not in the problem — the reflection, the foot of a perpendicular, the second intersection. Olympiad geometry lives on finding the right extra point.

Power of a point. For point P and circle Γ: PA·PB is constant for any line through P meeting Γ at A,B. This converts circles to products.

Inversion. Circles through the center become lines. Sometimes the inverted problem is trivial.

Angle chasing / cyclic quads. Four points are concyclic iff opposite angles sum to π. Chase angles until enough equalities force concyclicity.

Recurrence-specific trap

For recurrences like b_{n+1} = P(b_n) where P is polynomial degree ≥ 2: b_n grows doubly-exponentially. You cannot compute b_30 exactly — it has trillions of digits. Work in ℤ/2^m (or ℤ/p^m) from the start. Prove b_n ≡ r_n (mod 2^m) by induction on n, NOT by computing b_n.

When the answer involves √n or log n

These answers often come from a structure that is NOT the obvious/symmetric one. The diagonal, the identity, the "natural" choice frequently gives the WORST case, not the best — it clusters the constraint in a way that prevents large substructures.

For pure-reasoning solvers: Before claiming the symmetric choice is optimal, ask "what if I deliberately break the symmetry?" For grid/covering problems: what if the gaps are SPREAD OUT instead of clustered? For sequences: what if the extremal sequence is NOT constant or linear?

For deep-mode agents: Brute-force n=3..8 before theorizing. If the formula that fits is n+c√n instead of cn, the structure has √n-sized blocks.

The Look Back phase (after you have a proof)

• Can you check it? Plug in small cases. Does n=3 give what your formula says?
• Can you prove it differently? A second proof is a verification. And often shorter.
• Is your bound tight? If you proved ≤ N and the answer is exactly N, find the extremal case. If you can't, your bound might be loose.
• What did you actually use? Sometimes you used less than all the hypotheses — the real theorem is stronger.