# 2745. Construct the Longest New String

## Description

You are given three integers x, y, and z.

You have x strings equal to "AA", y strings equal to "BB", and z strings equal to "AB". You want to choose some (possibly all or none) of these strings and concatenate them in some order to form a new string. This new string must not contain "AAA" or "BBB" as a substring.

Return the maximum possible length of the new string.

A substring is a contiguous non-empty sequence of characters within a string.

Example 1:

Input: x = 2, y = 5, z = 1
Output: 12
Explanation: We can concactenate the strings "BB", "AA", "BB", "AA", "BB", and "AB" in that order. Then, our new string is "BBAABBAABBAB".
That string has length 12, and we can show that it is impossible to construct a string of longer length.


Example 2:

Input: x = 3, y = 2, z = 2
Output: 14
Explanation: We can concactenate the strings "AB", "AB", "AA", "BB", "AA", "BB", and "AA" in that order. Then, our new string is "ABABAABBAABBAA".
That string has length 14, and we can show that it is impossible to construct a string of longer length.


Constraints:

• 1 <= x, y, z <= 50

## Solutions

Solution 1: Case Discussion

We observe that the string ‘AA’ can only be followed by ‘BB’, and the string ‘AB’ can be placed at the beginning or end of the string. Therefore:

• If $x < y$, we can first alternately place ‘BBAABBAA..BB’, placing a total of $x$ ‘AA’ and $x+1$ ‘BB’, then place the remaining $z$ ‘AB’, with a total length of $(x \times 2 + z + 1) \times 2$;
• If $x > y$, we can first alternately place ‘AABBAABB..AA’, placing a total of $y$ ‘BB’ and $y+1$ ‘AA’, then place the remaining $z$ ‘AB’, with a total length of $(y \times 2 + z + 1) \times 2$;
• If $x = y$, we only need to alternately place ‘AABB’, placing a total of $x$ ‘AA’ and $y$ ‘BB’, then place the remaining $z$ ‘AB’, with a total length of $(x + y + z) \times 2$.

The time complexity is $O(1)$, and the space complexity is $O(1)$.

• class Solution {
public int longestString(int x, int y, int z) {
if (x < y) {
return (x * 2 + z + 1) * 2;
}
if (x > y) {
return (y * 2 + z + 1) * 2;
}
return (x + y + z) * 2;
}
}

• class Solution {
public:
int longestString(int x, int y, int z) {
if (x < y) {
return (x * 2 + z + 1) * 2;
}
if (x > y) {
return (y * 2 + z + 1) * 2;
}
return (x + y + z) * 2;
}
};

• class Solution:
def longestString(self, x: int, y: int, z: int) -> int:
if x < y:
return (x * 2 + z + 1) * 2
if x > y:
return (y * 2 + z + 1) * 2
return (x + y + z) * 2


• func longestString(x int, y int, z int) int {
if x < y {
return (x*2 + z + 1) * 2
}
if x > y {
return (y*2 + z + 1) * 2
}
return (x + y + z) * 2
}

• function longestString(x: number, y: number, z: number): number {
if (x < y) {
return (x * 2 + z + 1) * 2;
}
if (x > y) {
return (y * 2 + z + 1) * 2;
}
return (x + y + z) * 2;
}