365. Water and Jug Problem

You are given two jugs with capacities?x?and?y?litres. There is an infinite amount of water supply available. You need to determine whether it is possible to measure exactly?z?litres using these two jugs.

If?z?liters of water is measurable,you must have?z?liters of water contained within?one or both buckets?by the end.

Operations allowed:

• Fill any of the jugs completely with water.
• Empty any of the jugs.
• Pour water from one jug into another till the other jug is completely full or the first jug itself is empty.

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Example 1:?(From the famous?"Die Hard"?example)

Input: x = 3,y = 5,z = 4
Output: True

Example 2:

Input: x = 2,y = 6,z = 5
Output: False

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Approach #1: Math. [Java]

class Solution {
    public boolean canMeasureWater(int x,int y,int z) {
        if (x + y < z) return false;
        if (x == z || y == z || x + y == z) return true;
        return z % GCD(x,y) == 0;
    }
    
    private int GCD(int a,int b) {
        return b == 0 ? a : GCD(b,a % b);
    }
}

Analysis:

This is a pure Math problem. We need the knowledge of number theory to cover the proof and solution.?

The basic idea is to use the property of Bezout‘s identity and check is z is a multiple of GCD(x,y)

Quote from wiki:

Bezout‘s identity (also called Bezout‘s lemma) is a theorem in the elementary theory of numbers:

let a and b be nonzero integers and let d be their greatest common divisor. Then there exist integers x and y such that ax + by = d.

In addition,the greatest common divisord d is the smallest positive integer that can be written ad ax + by every integer of the form ax + by is a multiple of the freatest common divisor d.?

If a or b is negative this means we are emptying a jug of x or y gallons respectively.

Similarly if a or b is positive this means we are filling a jug of x or y gallons respectively.

x = 4,z = 8

GCD(4,6) = 2

8 is multiple of 2,so that this input is valid and we have:

-1 * 4 + 2 * 6 = 8

In this case,there is a solution obtained by filling the 6 gallon jug twice and emptying the 4 gallon jug once. (Solution. Fill the 6 gallon jug and empty 4 gallons to the 4 gallon jug. Empty the 4 gallon jug. Now empty the remaining two gallons from the 6 gallon jug to the 4 gallon jug. Next refill the 6 gallon jug. This gives 8 gallons in the end.)

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Reference:

https://leetcode.com/problems/water-and-jug-problem/discuss/83715/Math-solution-Java-solution