Solve Math Rigorously

Solve Math Rigorously

Overview

This skill is solver-first. Default to understanding the mathematical object, mapping the problem, choosing the right method family, solving in inspectable steps, and testing the result before sounding certain. If the source is a photo, screenshot, or uploaded page, first turn it into a trustworthy transcription and diagram map, then solve from that map. Teaching, worksheets, and lesson generation remain available, but they are secondary to accurate problem solving.

Claude-First Usage

This skill is intended to work well in Claude Desktop and Claude chat surfaces.

• Treat uploaded images and PDFs as first-class inputs.
• Prefer Claude's native multimodal reading first.
• Use the helper scripts only when the environment actually supports code execution.
• If code execution is unavailable, do not block on tooling. Continue with careful transcription, reasoning, and explicit uncertainty handling.

Read references/claude-surface-playbook.md before assuming tools or file-system access.

Default Solver Mode

Use this mode unless the user explicitly asks for something else.

Core standard:

1. Understand the problem before transforming it.
2. Build a problem map before committing to a method.
3. Solve one hinge step at a time.
4. Check every candidate result aggressively.
5. Present the logic in the order a strong human solver would want to read it.

Solver Operating Modes

• solver: full map, full method selection, clear derivation, and independent verification
• photo-solver: inspect the image, transcribe and map the task carefully, resolve or flag ambiguities, then solve and verify
• fast-answer: concise visible output, but still do the mapping and verification internally
• worked-solution: show the derivation in full, with the hinge step and method choice made explicit
• proof: separate exploration from proof, test for counterexamples early, then write a clean argument
• exam: minimize exposition, show only essential steps, and make the final answer easy to grade
• tutor: reveal the next move or next subgoal rather than dumping the whole solution at once

Secondary educational modes remain available when the user explicitly wants them:

• diagnostic
• worksheet
• lesson
• review
• primary-school
• parent-teacher

Photo Intake Protocol

If the task comes from a photo, screenshot, scanned page, uploaded PDF, whiteboard, or handwritten note:

1. Inspect the whole image before reading line by line. Determine whether it contains one problem, multiple problems, or a diagram plus text.
2. Delimit the task boundaries. Do not accidentally solve the wrong subproblem from the page.
3. Transcribe the problem exactly, preserving:
   • symbols
   • exponents
   • fractions
   • radicals
   • inequality signs
   • labels on diagrams, axes, and tables
4. Separate clearly read, inferred from context, and uncertain content.
5. For diagrams, record explicit givens separately from what only appears visually plausible. Do not infer exact equality or scale from a sketch unless it is marked or stated.
6. If legibility is poor and code execution is available, use scripts/math_photo_helper.py to create enhanced views before solving.
7. When ambiguity still matters after enhancement, state the competing readings and either ask the user to confirm or solve each plausible interpretation.

Read references/photo-mapping-playbook.md, references/diagram-reading-playbook.md, and references/claude-surface-playbook.md for photo-based tasks.

Problem Mapping Protocol

Before solving, map the task explicitly:

1. Restate the target in mathematical terms.
   • If the source is a photo, restate from the verified transcription, not from a guess.
2. Identify the mathematical object:
   • simplification or evaluation
   • equation or system
   • inequality
   • optimization
   • proof
   • counting or probability
   • recurrence or sequence
   • geometry or trigonometry
   • modeling or word problem
3. Define symbols, givens, unknowns, domain restrictions, units, hidden constraints, and any uncertain glyphs or labels coming from the source.
4. Decide what counts as a complete answer:
   • exact value
   • approximation
   • all solutions
   • proof
   • interval or region
   • extremum and where it occurs
5. List 2-3 plausible method families before choosing one.

Read references/problem-mapping-playbook.md when the problem is dense, ambiguous, or easy to misclassify.

Method Selection

Choose methods by triggers, not habit.

• Factor, substitute, or change form when the expression structure suggests it.
• Use symmetry, invariants, parity, or monotonicity when brute force looks wasteful.
• Use coordinates, vectors, or a diagram when geometry becomes algebra more cleanly than synthetic reasoning.
• Use derivative, convexity, or endpoint analysis for optimization.
• Use complements, conditioning, linearity of expectation, or counting models in probability and combinatorics.
• Use a small-case search or counterexample hunt before attempting a universal proof.

Read references/method-selection-playbook.md when choosing between competing approaches.

Solver Loop

Run the solve loop deliberately:

1. Choose the most promising branch.
2. Carry the branch until a hinge step succeeds or clearly stalls.
3. After each nontrivial step, checkpoint:
   • Is the transformation valid?
   • Did any domain restriction change?
   • Is the new form actually easier?
4. If the branch stalls, pivot instead of forcing it:
   • change representation
   • try a smaller case
   • isolate a subgoal
   • differentiate or factor
   • translate to coordinates or a table
5. Once a candidate answer appears, verify it before polishing the explanation.

Read references/solver-loop-playbook.md when the path is not obvious or when a first attempt stalls.

Step-By-Step Communication Standard

• Expose the hinge step. Do not skip the move that makes the solution work.
• If the source was a photo and the reading was not obvious, show the interpreted statement before solving.
• Justify the operation when it is not immediate.
• Keep notation stable. Do not silently rename objects or switch parameter meanings.
• Separate exact values from approximations.
• For proofs, separate exploration from the final proof.
• For systems or longer derivations, summarize subgoals as you move.
• If a result is only numerically supported, say that clearly.

Verification And Testing

Escalate verification until confidence is genuinely high:

1. Sanity checks:
   • sign
   • scale
   • units
   • symmetry
   • boundary behavior
   • special values
2. Structural checks:
   • substitute back
   • differentiate or integrate back
   • compare equivalent forms
   • verify constraints and excluded cases
3. Independent checks:
   • solve a second way
   • compute a small case
   • estimate numerically
   • use a geometric or probabilistic interpretation
4. Tool-assisted checks:
   • use scripts/math_verify.py
   • use scripts/math_visualize.py
   • use scripts/math_table.py when a table exposes the pattern better than prose
   • use scripts/math_photo_helper.py when better preprocessing is needed to trust the transcription

Read references/verification-playbook.md before claiming confidence on nontrivial work.

Tooling

If code execution is available and sympy, numpy, or matplotlib are unavailable, bootstrap a local environment from the skill root:

bash scripts/bootstrap_env.sh

Then run the helpers with .venv/bin/python3. If code execution is not available in the Claude surface, continue without scripts and make the verification status explicit in the response.

scripts/math_verify.py

Use this helper for direct solver-side checks.

Capabilities:

• expression equivalence
• derivative checks
• antiderivative checks
• definite integral checks
• substitution and evaluation
• direct relation satisfaction checks
• system satisfaction checks
• one-sided and two-sided limit checks
• finite solution-set comparisons
• counterexample search for equalities and inequalities on an interval

Examples:

.venv/bin/python3 scripts/math_verify.py equiv "sin(x)^2 + cos(x)^2" "1"
.venv/bin/python3 scripts/math_verify.py derivative "x^3 * exp(x)" "exp(x) * (x^3 + 3*x^2)" --var x
.venv/bin/python3 scripts/math_verify.py definite-integral "2*x*cos(x^2)" "0" "sqrt(pi)" "sin(pi)" --var x
.venv/bin/python3 scripts/math_verify.py solve "x^2 - 5*x + 6 = 0" --var x --expected 2 --expected 3
.venv/bin/python3 scripts/math_verify.py satisfies "x^2 + y^2 = 25" --assignment x=3 --assignment y=4
.venv/bin/python3 scripts/math_verify.py system --equation "x+y=5" --equation "x-y=1" --assignment x=3 --assignment y=2
.venv/bin/python3 scripts/math_verify.py limit "sin(x)/x" "0" "1" --var x
.venv/bin/python3 scripts/math_verify.py counterexample "sin(x) >= x" --var x --xmin 0.1 --xmax 2

Treat PASS as strong evidence. Treat FAIL as a real warning. Treat INCONCLUSIVE as a signal to add another check or change representations.

scripts/math_visualize.py

Use this helper when a plot or shaded figure makes the behavior easier to inspect:

.venv/bin/python3 scripts/math_visualize.py --expr "sin(x)" --expr "x/2" --xmin -6 --xmax 6 --output /tmp/sine-vs-line.png --title "Intersections of sin(x) and x/2"
.venv/bin/python3 scripts/math_visualize.py --expr "x^2" --shade-upper "x^2" --shade-lower "0" --shade-from 0 --shade-to 2 --xmin -1 --xmax 3 --output /tmp/area.png

scripts/math_table.py

Use this helper when a sequence, recurrence, or iteration is easier to inspect as a table:

.venv/bin/python3 scripts/math_table.py sequence "n^2 + n + 1" --start 1 --end 6
.venv/bin/python3 scripts/math_table.py iterate "cos(x)" --start 1 --steps 6

scripts/math_photo_helper.py

Use this helper when a math task comes from a raster image and the raw photo is hard to read cleanly, but only when the Claude surface supports code execution. For uploaded PDFs, prefer Claude's native PDF reading first.

Capabilities:

• upscale and normalize a photo
• produce grayscale, sharpened, and high-contrast variants
• split tall pages into readable tiles
• create a contact sheet for quick re-inspection

Examples:

.venv/bin/python3 scripts/math_photo_helper.py enhance --input /tmp/problem.jpg --output-dir /tmp/problem-enhanced
.venv/bin/python3 scripts/math_photo_helper.py enhance --input /tmp/worksheet.png --output-dir /tmp/worksheet-enhanced --upscale 2 --tile-height 1200

Use the enhanced outputs to verify the transcription before solving.

Secondary Educational Tools

Use these only when the user explicitly wants educational assets rather than solver-first output:

• scripts/math_manipulatives.py
• scripts/math_practice.py

scripts/run_validation_suite.py

Use this helper after changing solver tooling:

.venv/bin/python3 scripts/run_validation_suite.py

Solver References

Load these first for hard solving tasks:

• references/problem-mapping-playbook.md
• references/method-selection-playbook.md
• references/solver-loop-playbook.md
• references/claude-surface-playbook.md
• references/photo-mapping-playbook.md
• references/diagram-reading-playbook.md
• references/verification-playbook.md
• references/proof-playbook.md
• references/visualization-playbook.md

Load these only when the user explicitly wants teaching or educational materials:

• references/teaching-playbook.md
• references/primary-school-playbook.md
• references/diagnostic-playbook.md
• references/practice-playbook.md
• references/lesson-playbook.md
• references/mastery-playbook.md