# Puzzle Solutions, Issue 02

These solutions relate to puzzles found in Issue 02 of the magazine. Seven Digits Let’s call my number $n$. If the number is squared twice then multiplied by $n$, we get $n^5$. For all integers $n$, the final digit of $n^5$ is the same as the final digit of $n$. In other words: $$n^5\equiv n […] These solutions relate to puzzles found in Issue 02 of the magazine. ### Seven Digits Let’s call my number n. If the number is squared twice then multiplied by n, we get n^5. For all integers n, the final digit of n^5 is the same as the final digit of n. In other words:$$n^5\equiv n \mod 10$$Therefore, the final digit of Dr. Dingo’s number is 7.$$7^5=1680717^5=141985727^5=14348907$$So, in order for the answer to have seven digits, Dr. Dingo’s number was 17. ### Odd Squares If n^2 has all odd digits then the units digit of n must be odd. It can be checked that n cannot be a one digit number (except 1 or 3 as given in the question) as the tens digit will be even. Therefore n can be written as 10A+B where A is a positive integer and B is an odd positive integer.$$n^2=(10A+B)^2\\=100A+20AB+B^2
Now consider the tens digit of this.
$100A$ has no effect on this digit. The tens digit of $20AB$ will be the units digit of $2AB$ which will be even. The tens digit of $B^2$ is even (as checked above). Therefore the tens digit of $n^2$ is even.
Hence 1 and 9 are the only square numbers where all the digits are odd.

### Folding Tube Maps

Once the map is folded, it will look like this:

For the final tetrahedron to be regular, the red lengths must be equal. Let each red length be 2 (this will get rid of halves in the upcoming calculations). By drawing a vertical line in we can work out the width and height of the rectangle:

The width of the rectangle is 3 (one and a half red lengths). Using Pythagoras’ Theorem in the blue triangle, we find that the height of the rectangle is $\sqrt{3}$. Therefore, the ratio of the rectangle is $3:\sqrt{3}$ or $1:\sqrt{3}$.

The answers to these puzzles are available here.

Matthew is a postdoctoral researcher at University College London. He hasn’t had time to play Klax since the noughties, but he’s pretty sure that Coke is it!

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