Prison Cells After N Days - Solution & Explanation

This approach involves simulating each day's changes directly. Given the constraints of the problem, this can be manageable for relatively small values of 'n'. We iterate over each day, applying the rules to determine the state of each cell. Importantly, the edge cells are always vacant as they lack two neighbors. By updating the state iteratively, we can reach the final configuration after 'n' days.

The C program defines a function nextDay to calculate the state of cells for the next day based on the current state using the rules provided. The prisonAfterNDays function uses this helper function to simulate day-by-day changes. The logic accommodates the cyclic nature of the problem by using n % 14 to optimize large iterations, where 14 is the length of the repeating cycle determined by observation and testing.

This alternative approach builds on recognizing patterns within the transformations. Given only 8 cells and binary states, the maximum number of possible unique states is 2⁷ = 128. Through simulation, we can identify that the pattern cycles or repeats after at most 14 days, reducing the need to simulate all given 'n' days. By finding cycles, one can compute the state that would correspond to the final day directly, drastically minimizing operations.

The C code leverages a direct approach to identify patterns using the findCycle method, where state sequences are preserved and revisited to determine recurring configurations. This understanding is then used to restrict operations to only necessary repetitions through precise cycle length determination.