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Puzzle Solutions, Issue 04

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These solutions relate to puzzles found in Issue 04 of the magazine.

Odd sums

The sum of the first $n$ odd numbers is $n^2$ (this can be proved by induction). This means that:

$$\frac{\text{sum of the
first }n\text{ odd numbers}}{\text{sum of the next }n\text{ odd numbers}}=\frac{n^2}{(2n)^2-n^2}\\
=\frac{n^2}{3n^2}=\frac{1}{3}$$

Odd squares

1 and 9 are the only two square numbers whose digits are all odd.

Two lines

The co-ordinates will be (1,gradient+intercept).

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Matthew Scroggs is a PhD student at UCL working on finite and boundary element methods. His website, mscroggs.co.uk, is full of maths and now features a video of him completing a level of Pac-Man optimally.
Twitter  @mscroggs    Website  mscroggs.co.uk    + More articles by Matthew

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