Day 14 : Robot Redoubt

Link to Solution

Part 1: Simulating Robot Movement and Calculating the Safety Factor

Simulating Robot Movement:
The simulation begins by parsing the robot data, which includes the robots' initial positions and velocities. Each robot's data is represented as a tuple (p_x, p_y, v_x, v_y)—position and velocity components along the x and y axes.

The simulate function calculates the new positions of the robots after t seconds using the formula:

   p_x = (p_x + t * v_x) % width
   p_y = (p_y + t * v_y) % height

This formula accounts for the robot’s movement, updating its position at each time step and wrapping around the grid if it goes beyond the edges (due to the modulo operation). The robots are then placed back on the grid at the updated positions.

Quadrant Counting:
After simulating the robots at t = 100, the code counts the number of robots in each of the four quadrants of the grid. The grid is divided into quadrants based on the middle_row_gap and middle_column_gap, which are calculated as half the grid's width and height, respectively.

For each robot's position (x, y) after 100 seconds, the program checks which quadrant the robot occupies:

• Quadrant 0: Top-left
• Quadrant 1: Top-right
• Quadrant 2: Bottom-right
• Quadrant 3: Bottom-left

We then just get the product of the 4 quadrant's totals using Math.prod() function.

Part 2: Detecting the Christmas Tree Pattern

I made a few assumptions on this task, for example the image formed would be in the middle / centralised. As they're making a shape the robots must all be condensed together - forming the tree.

The robots move in predictable ways, and their positions can form specific shapes over time. To detect the "Christmas tree" pattern, the program looks for the time when the robots cluster into a tight formation that resembles the shape of a tree. The approach focuses on finding when the robots gather in a specific area of the grid.

The program starts by defining a large bounding box around all robots. This box is progressively reduced in size over time. The idea is that, as time passes, the robots will group together into a smaller region.

For each time step (each position of the robots), the program calculates how many robots are inside this shrinking box. It measures the density, which is the number of robots inside the box divided by the area of the box. The more robots inside the box, the higher the density.

The program tracks the time when the density is highest. When the density is at its maximum, the robots are most tightly packed, and this is likely when they form a recognisable shape (the Christmas tree).

Why does this work?

The method works because a "Christmas tree" pattern would cause the robots to cluster in a specific area of the grid. By shrinking the bounding box and calculating the density of robots in that area, the program can identify when the robots form this compact shape. The highest density indicates the robots are most tightly grouped, which corresponds to the Christmas tree formation.

Thus, the time step with the highest density is when the robots create the Christmas tree pattern.

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