given coordinates, find the number at that coordinates in spiral matrix. Given is the image of spiral i am talking about.
at 0,0 ---> 0
0,1 ---> 1
1,1 ---> 2
0,1 ---> 3
-1,1 ---> 4
-1,0 ---> 5
-1,-1 ---> 6
And the answer is here, but i don't know how to reach to this solution.
Somebody please give proof, how to reach to these equations.
Here's an attempted explanation of the first formula. It applies only in the open region to the right of the two lines $y=\pm x.$ If we show it is right when $y=0$ it will follow, since then the added $y$ will make it OK on the vertical line through $(x,0)$ while staying in the region.
A few sketches show that if we remove the points on the vertical line through $(x,0)$ with $y\le 0,$ what is left is a square of lattice points of side $2x-1.$ Then we need to add $x$ more, to account for the vertical part removed. But we also need to subtract one since the labeling starts at $0$ for the origin. This gives us $$(2x-1)^2+x-1=4x^2-3x,$$ which then on adding $y$ as noted to account for where one is on the last vertical part of the journey, gives the desired formula $4x^2-3x+y.$
The others are likely doable in a similar way, I'd suggest looking for a convenient point on an axis for a starting idea, and after some sketches finding a square (or maybe a rectangle near square) of lattice points which preceed your chosen axis point.