Think Like a Mathematician Task

ROLE: Act as an expert mathematician who narrates the REAL thinking process - including dead ends - so students learn how problem-solving actually feels.

CONTEXT: My gifted students in [CLASS/GRADE] (level [BOARD or EXAM e.g. RMO / JEE]) think 'real mathematicians just see the answer'. I want a task that shows the honest, messy process: exploring, getting stuck, trying things, recovering, then polishing.

YOUR TASK:
1. Take the problem I give (or pick one on [TOPIC]).
2. Write a 'THINKING OUT LOUD' narrative that models the process: first reactions, what to try first and why, at least one FALSE START and how you noticed it failed, the moment of insight, and how you recovered.
3. Then write the CLEAN final solution as it would appear in a textbook.
4. Reflect on the difference between the messy process and the clean writeup, and name 3 'mathematician habits' the narrative demonstrates (e.g. try small cases, seek symmetry, question assumptions).
5. Give 2 problems for students to practise the same habits.

INPUT: Here is the problem (or topic): [PASTE THE PROBLEM OR TOPIC HERE].

HOW TO WORK IT OUT: Genuinely solve the problem and let the narrative reflect plausible real reasoning, including a believable wrong turn that you then fix. VERIFY the final answer by two independent methods (e.g. an alternate approach plus a numerical or small-case check). The false start must be instructive, not nonsense, and you must clearly show why it failed. Never invent formulas or fake the recovery. Keep all mathematics fully legible in clean LaTeX or clear plain text.

OUTPUT FORMAT (exact headings, in order):
- THE PROBLEM (stated cleanly)
- THINKING OUT LOUD (the messy real process, with a false start and recovery)
- THE CLEAN SOLUTION (polished, textbook-style)
- FINAL ANSWER (clearly boxed)
- VERIFICATION (two independent checks shown)
- MESSY vs CLEAN (what the writeup hides)
- THREE MATHEMATICIAN HABITS ON SHOW
- PRACTISE THESE HABITS (2 problems)

RULES: The false start must be honest and instructive. Final answer double-checked. No bluffing in either the messy or clean version.

FORMAT THE OUTPUT — PRIMARY DELIVERABLE:
Return the complete material as a clean, print-ready A4 handout that can be pasted into Word or Google Docs and saved as PDF. Use a clear title, logical sections, readable mathematics and page-break markers where needed. Keep student material separate from the answer key or teacher notes.


MATHS FORMATTING: Write ALL mathematics so it reads correctly in a normal chat and pastes cleanly into Word or Google Docs. Use plain readable notation - for example a/b for fractions, x^2 for powers, sqrt(x) for roots, and symbols like <=, >=, !=, +-, pi, theta, integral and sum. Do NOT output raw LaTeX or code such as \frac{}{}, \sqrt{}, x^{2}, $...$ or \( ... \) - these appear as broken symbols to a teacher in a free chat.
At the very end add a footer line: "Prepared by [YOUR NAME]", and put "Compiled by [YOUR NAME]" in the header block.

CONTEXT CHECK: Before you begin, if any [SQUARE-BRACKET] above is still empty, or I have not given you the question / photo / data you need, ask me for exactly those and wait - never guess, and never answer with the placeholder text still in brackets.

GROUND RULES (always follow): Do not invent statistics, percentages, marks distributions you were not given, exam cut-offs, solve-time figures, NCERT or textbook page/line numbers, citations, or historical exam data. If you are unsure of any fact or number, say so and clearly label it an estimate; keep established fact separate from your own inference. Every step must be mathematically correct and at the stated class/board/exam level.

SIGNATURE: End your ENTIRE response with exactly this footer block (keep the lines and symbols):
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Questions, feedback or appreciation are always welcome.
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