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Home » a(2)^2 + b(2) + c &= 11, \\ - Groen Casting

a(2)^2 + b(2) + c &= 11, \\ - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 To solve this, we can use a combinatorial approach. We want to form an 8-digit number using the digits 3 and 4, with exactly three consecutive 3's. Let's break it down:

📌 Position the block of three consecutive 3's:

📌 The block "333" can start at positions 1 through 6 in the 8-digit number. This gives us 6 possible positions for the block.

📌 Fill the remaining 5 positions:

📌 After placing "333", we have 5 positions left. Each of these can be either a 3 or a 4, except that no additional block of three consecutive 3's is allowed. We need to ensure that no new block of three consecutive 3's is formed.

📌 Count valid configurations for each block position:

📌 If "333" starts at position 1:

📌 The sequence is "333xxxxx". The fourth digit cannot be 3 (to avoid "3333"), so it must be 4. The remaining 4 digits can be either 3 or 4, giving \(2^4 = 16\) configurations.

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