Can You See The Forest From The Trees?

Interesting problem!

The number of trees we see between due East and due North East (including) is equal to the number of distinct rational numbers on the interval $(0,1]$ with denominator $\leq 50$ , which is $\sum_{k=1}^{50}\phi(k)=774$ . Thus the total number will be $8\times 774=\boxed{6192}$ .

Interesting problem.

I plotted the lower right quadrant of the forest just for interest. I find it interesting how distinctive the prime number appear in this plot. It seams to me that there could likely be other problems that could be imagined with regard to the distribution of the numbers in the solution to this problem. I have considered posting a similar question. Sticking with the forest of trees description, One could stipulate some areas with no trees. perhaps a clear cut area with a given radius around the viewer, clear cut areas of different shapes placed somewhere in the forest, clear cut grid of roads removing entire columns or entire rows in the forest.

Let the center be $(0,0)$ .

A tree $(x_1, y_1)$ is hidden iff there exists another tree $(x_2, y_2)$ such that $x_1 = kx_2, y_1 = ky_2$ , where $k \ge 2$ is a positive integer. In other words, a tree is visible iff the GCD of their coordinates is 1. Now it's straightforward:

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>>> sum((math.gcd(x,y) == 1) for x in range(-50,51) for y in range(-50,51))
6192

This is generalized easily to higher dimensions: a lattice point is visible from the origin iff the GCD of all their coordinates is 1.

Call the central position $(0, 0)$ , then the forest corresponds to the set $\left\{(x, y)|-50 \leq x,y \leq 50\right\},$ and you can see a tree precisely if the ratio $x:y$ cannot be reduced to smaller terms, i.e. if $\text{gcd}(|x|,|y|) = 1.$ The implementation is straightforward:

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int trees = 0, size = 50;

for (int x = -size; x <= size; x++)
    for (int y = -size; y <= size; y++) if (x != 0 || y != 0) 
    {
        int a = (x < 0 ? -x : x), b = (y < 0 ? -y : y);
        while (b > 0) {
            int c = b; b = a % b; a = c;    // i.e. a = gcd(|x|, |y|);
        }
        if (a == 1) trees++;
    }

out.println(trees);

Output:

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6192

Note : In this case $6192/10200 = 60.706\%$ of the trees are visible As the forest grows bigger, the fraction of visible trees approaches the odds that two arbitrarily chosen integers are coprime. This limiting probability is $6/\pi^2 \approx 60.793\%$ .

just for the record, this is the sequence of coprime pairs, OEIS 137243, term 50.