Velvet Star Monitor

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Prove that there is only one way to make a square using all seven tangram pieces

I am pretty sure there is only one way to make a square from the seven tangram pieces: How can I prove this is the only way respecting all symmetries?

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If $\tan(\theta$) = $\frac{x}{7}$ for -$\frac{\pi}{2} < \theta < \frac{\pi}{2}$, find an expression for $\sin(2\theta$) in terms of $x$.

I'm trying to solve this and am getting a very different answer from the book. Can someone please help. I know that $\sin(2\theta) = 2\sin(\theta)\co...

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How many Chia plots can I fit on a hard drive, when attempting to fill it completely and maximizing the overall number of plots?

I'm making plots for Chia, a cryptocurrency which isn't "mined" in the strict sense of the word, but instead requires that one have ma...

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Spectral theorem of compact operators in Hilbert space

I am reading the following theorem from my lecture notes (English translation of German text). But I don't understand exactly what is meant from this...

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LCD of 2x+1, x^2 and x

I am given the following sum: $$\frac{x}{2x+1} + \frac{3}{x^2} + \frac{1}{x}$$ In order to add these fractions, I must find a common denominator. I have b...

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Number of $4$-letter "words'' that can be constructed with the $12$-letter Hawaiian alphabet

The Hawaiian alphabet consists of $12$ letters, the vowels a, e, i, o, u and the consonants h, k, l, m, n, p, w. Show that $20,736$ different $4$-letter ‘...

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Most important Linear Algebra theorems?

I was reading up on symmetric matrices and the textbook noted that the following is a remarkable theorem: A matrix $A$ is orthogonally diagonalizable iff ...

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when product of irrational numbers = rational number?

let $a$ and $b$ be irrational numbers. when do we have $ a \cdot b $ = rational number? for example $\sqrt{2} \cdot \sqrt{2}=2$. I was wondering if there ...

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Calculate fillet radius

I want to create a fillet from a predefined point p of an arc A1 to a second arc A2. I need to calculate the radius f. I have 3 cases where it needs to be...

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How do you prove arc length is greater than chord length?

Graphically, it's obvious that given two different points $a$ and $b$ on a circle of radius $r$, the linear distance (chord length) from $a$ to $b$ i...

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