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Home » \textTotal ways = \binom154 = 1365 - ToelettAPP

\textTotal ways = \binom154 = 1365 - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Let’s find $(x, y, z)$ such that its distance to each of the three points is also $\sqrt{2}$:

📌 Distance from $(0,0,0)$:

📌 x^2 + y^2 + z^2 = 2

📌 Distance from $(1,1,0)$:

📌 \Rightarrow x^2 - 2x + 1 + y^2 - 2y + 1 + z^2 = 2

📌 \Rightarrow (x^2 + y^2 + z^2) - 2x - 2y + 2 = 2

📌 \Rightarrow 2 - 2x - 2y + 2 = 2 \Rightarrow -2x - 2y = -2 \Rightarrow x + y = 1

📌 Distance from $(1,0,1)$:

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R(1) &= 3(1)^2 + 2(1) + 1 = 3 + 2 + 1 = 6, \\
R(2) &= 3(4) + 4 + 1 = 12 + 4 + 1 = 17, \\
R(3) &= 3(9) + 6 + 1 = 27 + 6 + 1 = 34, \\
R(4) &= 3(16) + 8 + 1 = 48 + 8 + 1 = 57, \\
R(5) &= 3(25) + 10 + 1 = 75 + 10 + 1 = 86, \\
R(6) &= 3(36) + 12 + 1 = 108 + 12 + 1 = 121, \\
R(7) &= 3(49) + 14 + 1 = 147 + 14 + 1 = 162, \\
R(8) &= 3(64) + 16 + 1 = 192 + 16 + 1 = 209, \\
R(9) &= 3(81) + 18 + 1 = 243 + 18 + 1 = 262, \\
R(10) &= 3(100) + 20 + 1 = 300 + 20 + 1 = 321.

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