posted on 20180320
Recall that the symbol $\mathbb{Z}^2$ denotes the set of all coordinate pairs with integer components.
A spiral of length $n\ge 1$ is a counterclockwise spiral starting at $(0,0)$ extending to the right to $(n,0)$, wrapping around so as to cover $\mathbb{Z}^2$ completely.
Consider the following spiral of length $2$.
******
 
* **** *
  (0,0)  
* * x*x *
  (2,0) 
* *****

****** ...
The points of the spiral have a natural ordering. The above spiral can be written as a sequence $s$ where $$ \begin{align} s_0 &= (0,0), & s_3 &= (2,1),\\ s_1 &= (1,0), & s_4 &= (1,1),\\ s_2 &= (2,0), & s_5 &= (0,1), \end{align} $$ and so on.
There are three challenges. Given an $n\ge 1$, do the following.
Given a $k\ge 0$, find $s_k$.
Given a point $(\alpha, \beta)\in\mathbb{Z}^2$, find the $k$ such that $s_k = (\alpha,\beta)$.
Print the spiral of length $2$ to the terminal by printing the values of $k$ in their correct positions. (And make it look nice by ensuring the numbers are aligned and reasonably spaced out.)