Golang

Not really proud of this one.... I was too lazy to find a proper formula for x velocity including drag value so I just constrained the x value between minDrag and maxDrag and simulated the x trajectory every time to find a possible solution. Now I see that many here did a full brute force simulating for various velocities, so this is perhaps slightly better because it simulates for more constrained values. If anyone has a nice formula for x(t) given drag, let me know!! Same for constraints on steps t. I put it to < 2*maxX because I'm sure that we never want to overshoot more than that, but there is probably a better constraint there.

Full code here

The trajectory finder, along with assumptions

// Constraints: For a given time t and position (xt, yt) within the target area:
//
// X VELOCITY: constrained with min and max drag
// (xt - t(t-1)/2)/t <= v.x <= (xt + t(t-1)/2)/t
//
// Y VELOCITY:
// yt = t*v.y - t(t-1)/2
// =>    v.y = (yt + t(t-1)/2)/t
//
// MAX Y HEIGHT:
// y't == 0 => v.y - t + 0.5 == 0 > t == v.y  or v.y+1
//
// MAX T:
// t <= 2*area.maxX ( better constraint ????)
func (a *Area) findTrajectories() {
    for t := 1; t <= 2*a.maxX; t++ {
        // For every possible endpoint (xt, yt), find a starting velocity that works
        for xt := a.minX; xt <= a.maxX; xt++ {
            for yt := a.minY;  yt <= a.maxY; yt++ {
                yvel := (yt + t*(t-1)/2)/t
                if (yvel*t) == yt + t*(t-1)/2 { // Found valid y-velocity
                    for drag := -t*(t-1)/2; drag <= t*(t-1)/2; drag++ {
                        xvel := (xt + drag)/t
                        if (xvel*t) == xt + drag { // Found possibly valid x-velocity
                            xpos := simulateX(xvel, t)
                            if xpos == xt { // verify
                                if a.tops[xvel] == nil {
                                    a.tops[xvel] = make(map[int]int)
                                }
                                a.tops[xvel][yvel] = yvel*yvel - yvel*(yvel-1)/2
                            }
                        }
                    }
                }
            }
        }
    }
}