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# Sunday Afternoon Maths VIII

Posted on 2014-04-13

## Rebounds

In a 4x3 rectangle, a ball is fired from the top left corner at 45°.
It bounces around a rectangle until it hits a corner. Which corner does it end in?
Which corner will it end in for rectangles of other sizes?
Tags: geometry

## Complex squares

For which complex numbers, $$z$$, are $$\mathrm{Re}(z^2)$$ and $$\mathrm{Im}(z^2)$$ both positive?

Let $$a_b$$ denote $$a$$ in base $$b$$.
Find bases $$A$$, $$B$$ and $$C$$ less than 10 such that $$12_A+34_B=56_C$$.
Tags: numbers, bases

## Reverse bases again

Find three digits $$a$$, $$b$$ and $$c$$ such that $$abc$$ in base 10 is equal to $$cba$$ in base 9?
Tags: numbers, bases

## Two

Find $$a$$ such that $$a+(a+A)^{-1}=2$$, where $$A=(a+A)^{-1}$$.
ie. $$a + \frac{1}{a + \frac{1}{a + \frac{1}{a + \frac{1}{...}}}} = 2$$.
Find $$b$$ such that $$b+(b+B)^{\frac{1}{2}}=2$$, where $$B=(b+B)^{\frac{1}{2}}$$.
ie. $$b + \sqrt{b + \sqrt{b + \sqrt{b + \sqrt{...}}}} = 2$$.
Find $$c$$ such that $$c+(c+C)^{2}=2$$, where $$C=(c+C)^{2}$$.
In terms of $$k$$, find $$d$$ such that $$d+(d+D)^{k}=2$$, where $$D=(d+D)^{k}$$.
Tags: numbers
If you enjoyed these puzzles, check out Advent calendar 2019,
puzzles about triangle numbers, or a random puzzle.

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