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Home » Discover the Tongue-Twisting Truth: What’s the LARGEST Horse Breed Ever? - Abbey Badges

Discover the Tongue-Twisting Truth: What’s the LARGEST Horse Breed Ever? - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Start with the equation:

📌 \cos 2\theta = \sin \theta

📌 Use the double-angle identity for cosine:

📌 \cos 2\theta = 1 - 2\sin^2\theta

📌 - 2\sin^2\theta = \sin \theta

📌 Rearrange into a quadratic in \(\sin \theta\):

📌 2\sin^2\theta + \sin \theta - 1 = 0

📌 Let \( x = \sin \theta \), so:

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But for small $n=8$, $k=3$, we use a constructive method.
Fix one station as selected. By rotational symmetry, fix station 1. Th...
So available: 3, 4, 5, 6, 7, excluding 2 and 8 → valid: 3–7.
We need 2 non-adjacent from {3,4,5,6,7}.
List all 2-element non-adjacent pairs in 5 consecutive positions:
Total: 6 valid pairs with station 1 selected.
But each selection is counted 3 times (once for each selected station)...
\frac{8 \times 6}{3} = 16
Thus, number of non-adjacent triplets is 16.
\frac{16}{56} = \frac{2}{7}

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