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Home » \(\sqrt345 \approx 18.57\), so - Anadea

\(\sqrt345 \approx 18.57\), so - Anadea

📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 Alternatively, note symmetry. For uniformly random $ a < b < c $, we analyze $ 2b - a - c $. We want $ 2b < a + c $.

📌 Then $ 2b < a + c = (b - d_1) + (b + d_2) = 2b + (d_2 - d_1) $, so:

📌 Thus, the condition $ 2b < a + c $ is equivalent to $ c - b > b - a $, i.e., the gap from $ b $ to $ c $ is larger than to $ a $.

📌 So for increasing triples $ a < b < c $, with spacing $ d_1 = b - a $, $ d_2 = c - b $, we require $ d_2 > d_1 $.

📌 We fix $ b $, and for each $ d_1 = 1 $ to $ b - 1 $, $ d_2 = d_1 + 1 $ to $ 8 - b $, count valid $ (d_1, d_2) $, then sum.

📌 Each corresponds to exactly one ordered triple with $ a < b < c $ satisfying the condition.

📌 Question: A topologist labels 6 distinct regions of a cell membrane using 3 red and 3 blue topological markers, assigning one marker per region uniformly at random. What is the probability that no two adjacent regions (in a fixed linear chain) are both colored red?

📌 This is a classic non-adjacent independent set problem.

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