365. Water and Jug Problem

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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.

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

This is a pure Math problem. We need the knowledge of number theory to cover the proof and solution. No idea why microsoft uses this problem in real interview.

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

public boolean canMeasureWater(int x, int y, int z) {
    //limit brought by the statement that water is finallly in one or both buckets
    if(x + y < z) return false;
    //case x or y is zero
    if( x == z || y == z || x + y == z ) return true;
    
    //get GCD, then we can use the property of Bézout's identity
    return z%GCD(x, y) == 0;
}

public int GCD(int a, int b){
    while(b != 0 ){
        int temp = b;
        b = a%b;
        a = temp;
    }
    return a;
}