World Challenge

The Orb Operator

The world is a \(20 \times 20\) grid of 400 towns with the origin at the bottom left. Each town has coordinates \(t = (x,y)\) with \(x,y \in \{0, \ldots, 19\}\). The population of town \(t\) is

\(P_{x,y} = 50 + \lfloor 20\sin(0.35x) + 15\cos(0.30y) + 10\sin(0.20(x+y)) \rfloor.\)

(All trigonometric functions use radians.)

Every town has a fixed selfie rate \(s_t\), representing the fraction of people who get distracted taking pictures with the Orb instead of actually verifying: \(s_t = 0.098765432\) for the four corners \((0,0), (0,19), (19,0), (19,19)\), \(s_t = 0.135792468\) for non-corner boundary towns, and \(s_t = 0.246913579\) for all interior towns. The effective participation rate at town \(t\) is \(1-s_t\).

There are 10 Orbs placed at continuous coordinates \((x_i, y_i)\) inside the square \([0,19]^2\), with the decentralization rule that the Euclidean distance between any two Orbs must be at least 2.5. The distance from a town \((x,y)\) to an Orb \((x_i, y_i)\) is

\(d((x,y), (x_i, y_i)) = \sqrt{(x-x_i)^2 + (y-y_i)^2}\)

and each town is assigned to its nearest Orb. If a town is served at distance \(d\), each person makes exactly one verification attempt with probability \(\alpha(d) = 0.50e^{-0.40d}\).

The number of successful verifications from each town depends on the population, selfie rate, and the distance to the nearest Orb. Your goal is to position the Orbs to maximize the total expected number of successful verifications across all 400 towns.

Submit the coordinates of your 10 Orbs to maximize expected successful verifications: [(x1,y1), (x2,y2), ..., (x10,y10)]