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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?

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Tags: geometry

Complex squares

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

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Adding bases

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

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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?

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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 \(D=(d+D)^{k}\).

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Tags: numbers
If you enjoyed these puzzles, check out Sunday Afternoon Maths LXVII,
puzzles about star numbers, or a random puzzle.

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