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Home » How I Permanently Silenced My Rival Sister—The Shocking One-Hit Payoff! - Upplift

How I Permanently Silenced My Rival Sister—The Shocking One-Hit Payoff! - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 1 block of discoveries from the 18th century (4 discoveries)

📌 1 block of discoveries from the 19th century (2 discoveries)

📌 First, we calculate the number of ways to arrange the three blocks. Since there are 3 blocks, the number of arrangements is:

📌 Next, within each block, the discoveries can be arranged among themselves:

📌 The 3 discoveries in the 17th century block can be arranged in \(3! = 6\) ways.

📌 The 4 discoveries in the 18th century block can be arranged in \(4! = 24\) ways.

📌 The 2 discoveries in the 19th century block can be arranged in \(2! = 2\) ways.

📌 Therefore, the total number of distinct arrangements is the product of the arrangements of the blocks and the arrangements within each block:

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So possible coordinates: $(0,1,1)$ or $(0,-1,-1)$.
Check distances:
From $(0,1,1)$ to $(0,0,0)$: $\sqrt{0 + 1 + 1} = \sqrt{2}$
To $(1,1,0)$: $\sqrt{1 + 0 + 1} = \sqrt{2}$
To $(1,0,1)$: $\sqrt{1 + 1 + 0} = \sqrt{2}$
All distances match.
Thus, the fourth vertex is $\boxed{(0,1,1)}$.
Question: Find all angles $z \in [0^\circ, 360^\circ]$ that satisfy
2\sin^2 z - 3\sin z + 1 = 0
Let $x = \sin z$. Then:

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