arrangingCoins.ts

/*

You have n coins and you want to build a staircase with these coins.
The staircase consists of k rows where the ith row has exactly i coins. The last row of the staircase may be incomplete.
Given the integer n, return the number of complete rows of the staircase you will build.

Example 1:
Input: n = 5
Output: 2
Explanation: Because the 3rd row is incomplete, we return 2.


Example 2:
Input: n = 8
Output: 3
Explanation: Because the 4th row is incomplete, we return 3.

*/

function arrangeCoins(n: number): number {
	// At the kth rows, we will have the maximum total is k*(k+1) coins
	// So to find the rows, we just need to find the number k that satisfies the condition k*(k+1)/2 >= n
	// If k*(k+1)/2 = n, then return k
	// And if k*(k+1)/2 > n, then return k - 1, because row kth is not completed
	// We can use binary search since the k in the array from 1 to n (it is sorted array)
	// Time complexity: O(log(n))
	// Space complexity: O(1)
	let left = 1;
	let right = n;

	while (left <= right) {
		let mid = Math.floor((left + right) / 2);
		let temp = (mid * (mid + 1)) / 2;

		if (temp === n) {
			return mid;
		}

		if (temp < n) {
			left = mid + 1;
		} else {
			right = mid - 1;
		}
	}

	return left - 1;

	// Approach 2
	// Using math for calculate
	// k * (k + 1) / 2 <= n (n > 0)
	// <=> k^2 + k - 2n <= 0
	// <=> k <= (-1 + √(1 + 8n)) / 2 (Using the quadratic formula k = (-b ± √(b² - 4ac)) / 2a, where a = 1, b = 1, and c = -2n)
	// return Math.floor((-1 + Math.sqrt(1 + 8 * n)) / 2);
}

console.log(arrangeCoins(1));