Hard Problem with a Hints Ladder

ROLE: Act as a master tutor who designs a graded HINT LADDER so students solve a hard problem mostly by themselves.

CONTEXT: My gifted students in [CLASS/GRADE] (level [BOARD or EXAM e.g. RMO / JEE Advanced]) give up too fast on hard problems. I want one tough problem with hints that reveal the idea in small, increasing doses, so they struggle productively.

YOUR TASK:
1. Take the hard problem I give (or pick a suitable one on [TOPIC]).
2. Design a ladder of 5 hints, from a gentle nudge to almost-the-answer: Hint 1 = direction only; Hint 5 = the key step spelled out, but NOT the full solution.
3. Each hint must be useful on its own and must not leap past the previous one.
4. After the ladder, give the COMPLETE solution and final answer.
5. Add a one-line note on the single insight that unlocks the problem.

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

HOW TO WORK IT OUT: Solve the problem fully yourself first, then design hints backwards from the real solution so each hint is genuinely on the solution path. VERIFY the final answer by two independent methods (e.g. an alternate route plus substitution or a small-case/numerical check). Never write a hint that points the wrong way, and never invent a step. Keep all mathematics fully legible in clean LaTeX or clear plain text. Order the hints from least to most revealing and keep that ordering strict.

OUTPUT FORMAT (exact headings, in order):
- THE PROBLEM (stated cleanly)
- HINT LADDER: HINT 1 (gentlest) ... HINT 5 (key step) - each clearly separated so I can reveal one at a time
- FULL SOLUTION (step by step)
- FINAL ANSWER (clearly boxed)
- VERIFICATION (two independent checks shown)
- THE UNLOCKING INSIGHT (one line)

RULES: Hints must be strictly increasing in helpfulness and never reveal the full solution before the solution section. Answer double-checked. No bluffing.

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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🔗 Free tools & to reach me: https://jee-prompt-studio.pages.dev/
Questions, feedback or appreciation are always welcome.
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