How To Solving the Equation: (sqrt(1 + 8 * long(num)) – 1) / 2
LeetCode, a popular platform for coding challenges, often presents intriguing problems that require creative solutions. Today, we’re diving into a problem that involves stacking coins. The task is to determine how many stacks of coins you can form when you have a certain number of coins, where each row has a different number of coins. This problem can be challenging, but we have a formula that can help us crack it. We’ll explore the formula (sqrt(1 + 8 * long(num)) – 1) / 2 and understand how it works.
The Problem Statement:
Before we delve into the formula, let’s first understand the problem statement. The question posed on LeetCode is as follows: “How many stacks of coins can you form with ‘n’ number of coins, where the k-th row has ‘k’ number of coins?” You can check out the problem on LeetCode’s website under the July Leetcoding Challenge, Week 1.
The Formula:
The formula for solving this problem is (sqrt(1 + 8 * long(num)) – 1) / 2. It may appear complex at first glance, but it has a simple explanation. To understand how this formula works, we need to break it down step by step.
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The Sum of the First N Natural Numbers:
First, let’s revisit the mathematical concept of the sum of the first ‘N’ natural numbers, which is often written as 1 + 2 + 3 + … + N. This sum is known to be equal to N(N+1)/2. In the context of stacking coins, this sum represents the total number of coins required to form ‘N’ rows.
- Stacking Coins:
The task in the problem is to stack coins in a specific way. For example, if you have 6 coins, you should stack them as follows:
- x (1 coin)
- x x (2 coins)
- x x x (3 coins)
The number of coins in each row corresponds to the sequence of natural numbers (1, 2, 3). In this case, 6 is equal to 1 + 2 + 3. This relationship between the total number of coins and the sum of natural numbers forms the basis of our solution.
- Finding ‘N’ from ‘K’:
Now, let’s address the main question: Given ‘K’ coins, how can we find the value of ‘N’? We start with the equation:
N(N+1)/2 = K
Solving for ‘N’, we get:
N^2 + N = 2K N^2 + N – 2K = 0
Using the quadratic formula, we can find ‘N’:
N = (-1 + sqrt(1 + 8K)) / 2 N = -1/2 + sqrt(1/4 + 2K) N = sqrt(2K + 0.25) – 0.5
Conclusion:
In conclusion, the formula (sqrt(1 + 8 * long(num)) – 1) / 2 is a clever way to find the number of stacks of coins you can form when given ‘K’ coins. It is based on the sum of natural numbers and provides a quick solution to the problem posed on LeetCode.
We hope this explanation helps you understand the solution better and empowers you to tackle similar problems in the future. Happy coding!
