## Solution

#### Approach 1: Simulation

Intuition

We simulate each day of the prison.

Because there are at most 256 possible states for the prison, eventually the states repeat into a cycle rather quickly. We can keep track of when the states repeat to find the period t of this cycle, and skip days in multiples of t.

Algorithm

Let's do a naive simulation, iterating one day at a time. For each day, we will decrement N, the number of days remaining, and transform the state of the prison forward (state -> nextDay(state)).

If we reach a state we have seen before, we know how many days ago it occurred, say t. Then, because of this cycle, we can do N %= t. This ensures that our algorithm only needs steps.

Complexity Analysis

• Time Complexity: , where is the number of cells in the prison.

• Space Complexity: .

Analysis written by: @awice.