MHB Securing n Sheets with Thumbtacks: Can You Prove My Conjecture?”

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The discussion revolves around securing an infinite number of square sheets of paper on a bulletin board using thumbtacks, with the requirement that all four corners of each sheet must be secured. The key point is that overlapping sheets allows one thumbtack to secure multiple sheets simultaneously, leading to a reduction in the total number of tacks needed. The conjecture presented suggests a specific formula for calculating the minimum number of thumbtacks required for n sheets, with examples indicating a pattern in the number of tacks needed as more sheets are added. Participants are encouraged to explore ways to prove this conjecture mathematically. The conversation emphasizes the challenge of deriving a formal proof for the proposed formula.
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You have an infinite supply of square sheets of paper. You are going to secure these sheets on an infinitely large bulletin board by using thumbtacks. You must secure all four corners of each sheet however you may slightly overlap the sheets so that one thumbtack could secure up to four sheets at once. Under these assumptions, one sheet requires 4 tacks, 2 sheets require 6 tacks, 3 sheets require 8 tacks, etc.

What is the minimum number of thumbtacks needed to secure $$n$$ sheets?

I have a conjecture for a formula but have no clue how to prove it.
 
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Here is my conjecture for the number of thumbtacks needed for $$n$$ sheets of paper.

$$T(n) = \lceil(1+\sqrt{n})^2 \rceil$$ where the upper brackets represent the ceiling function.
 
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