Count Distinct Numbers on Board - Solution & Explanation

This approach works by reverse engineering the board filling process. Given a number n, we find the numbers that can potentially go on the board. These numbers are such that x % i == 1 for each previously existing number x on the board. Through observation, you'll notice that once a number is included on the board, it allows for smaller numbers to be generated. By understanding the modulo behavior, we can establish that the smallest sequence achievable sorts to n, n-1, ..., 2 after limitless days.

The C solution uses a simple function that takes n as an input and returns n-1. This solution works because the process fills out all possible numbers such that 1 <= i < n. For n=5, these are 2, 3, 4.

In this approach, we simulate the process of adding numbers to the board for small values of n to identify a pattern. By listing the numbers that result from x % i == 1, it becomes evident that the upper bound determines the potential set of numbers you can derive. This results in numbers from 2 to n eventually appearing on the board over time.

This C implementation uses a simulation to justify a similar conclusion. By observing patterns from smaller simulations, it simplifies this to the return statement n - 1.