# Background

The idea of this article originated from a contest (Petrozavodsk Summer-2016. Petr Mitrichev Contest 14), which I believe is attributed to Petr. In this contest, an interesting problem is proposed:

"Cosider this process: pick a random number *n*_{i} uniformly at random between 10 and 100. Generate *n*_{i} random points with integer coordinates, picking each coordinate independently and uniformly at random from all integers between 0 and 10^{9}, inclusive. Find the convex hull of those points.

Now you are given 10000 polygons generated by this program. For each polygon, you need to guess the value *n*_{i} that was used for generating it.

Your answer will be accepted if the average (over all 10000 hulls) absolute difference between the natural logarithm of your guess and the natural logarithm of the true *n*_{i} is below 0.2."

Unfortunately, I didn't really manage to work this one out during our 5-hour training session. After the training is over, however, I have tried to read the solution program written by Petr, which looks like the following:

```
//...
public class h {
static int[] splitBy = new int[] {/* 1000 seemingly-random elements */};
static double[] splitVal = new double[] {/* another 1000 seemingly-arbitrarily-chosen elements */};
static double[] adjYes = new double[] {/* Another 1000 seemingly-stochastically-generated elements */};
static double[] adjNo = new double[] {/* ANOTHER 1000 seemingly-... elements, I'm really at my wit's end */};
public static void main(String[] args) {
/* Process the convex hull, so that
key.data[0] is the average length of the convex hull to four sides of the square border
(i.e. (0, 0) - (1E9, 1E9));
key.data[1] is the area of the hull;
key.data[2] is the number of points on the hull.
*/
double res = 0;
for (int ti = 0; ti < splitBy.length; ++ti) {
if (key.data[splitBy[ti]] >= splitVal[ti]) {
res += adjYes[ti];
} else {
res += adjNo[ti];
}
}
int guess = (int) Math.round (res);
if (guess < 10) guess = 10;
if (guess > 100) guess = 100;
pw.println (guess);
}
}
```

While I was struggling to understand where all the "magic numbers" come from, I do realize that the whole program is somewhat akin to a "features to output" black box, which is extensively studied in machine learning. So, I have made my own attempt at building a learner that can solve the above problem.

# A lightweight learner

Apparently, most online judge simply do not support scikit-learn or tensorflow, which are common machine learning libraries in Python. (Or an 100MB model file, the imagination of 500 users with an 100MB file each makes my head ache. And yes, there are even multiple submissions.) Therefore, some handcraft code is necessary to implement a learner that is easy to use.

As a university student, I do not know much about machine learning, especially in regression, where even fewer methods are adaptable. However, I somehow got particularly attracted by the idea of the neural network after some googling, both because its simplicity and its wide application.