Water and Jug Problem

Problem Description

You are given two jugs with capacities x liters and y liters. You have an infinite water supply. Return whether the total amount of water in both jugs may reach target using the following operations:

• Fill either jug completely with water.
• Completely empty either jug.
• Pour water from one jug into another until the receiving jug is full, or the transferring jug is empty.

Key Insights

• The problem can be reduced to checking if the target amount of water can be expressed as a linear combination of the capacities of the two jugs.
• The target amount must not exceed the total capacity of both jugs combined.
• The greatest common divisor (GCD) of the two jug capacities plays a crucial role: the target must be a multiple of the GCD of the two capacities.

Space and Time Complexity

Time Complexity: O(log(min(x, y))) - for calculating the GCD Space Complexity: O(1) - no additional space is used other than variables

Solution

To solve this problem, we can use the mathematical properties of the GCD. Specifically, we check the following:

1. If the target is greater than the sum of the two jug capacities, return false.
2. If the target is a multiple of the GCD of the two jug capacities, return true; otherwise, return false.

This approach leverages the mathematical properties of integers and their combinations, ensuring an efficient solution.

Code Solutions