# Orchard Planting: Greatest Common Divisor

The Orchard Planting contest from infinite search space is over. So it is time for a quick write-up.

The rules are simple, on a grid of integers, place N points on the grid to get as much 4 points on a line and never more then 4 points on a line.

My big break-through was when I figured out a way to improve the calculation speed of a solution, and make it possible to extend existing solutions (going back and forwards). To do this I used a unique vector (greatest common divisor vector) which is the same for all point on the same line:

Now we can evaluate the points:

If a point has three vectors that are the same, we have a line with four points! This can be checked easily if you sort the vectors and go through them once.

Also adding and removing points becomes very easy. A lot of the GCD calculations can be cached. To remove a point, just remove the vectors it made. And to add a point, calculate all the new vectors. So in the end it basically all boils down to a lot of GCD calculations and sorting.

Was this the fastest way to calculate solutions in this contest? I don’t know, but I was really pleased when I figured it out. With a better algorithm for picking possible numbers (instead of hill-climbing) and some more processor power I bet I could have ended a bit higher up the hill.

Also: Keep an eye out for the next contest, it is going to be an interesting one! January the 13th.