# leetcode\_1041

On an infinite plane, a robot initially stands at (0, 0) and faces north. The robot can receive one of three instructions:

"G": go straight 1 unit; "L": turn 90 degrees to the left; "R": turn 90 degress to the right. The robot performs the instructions given in order, and repeats them forever.

Return true if and only if there exists a circle in the plane such that the robot never leaves the circle.

Example 1:

Input: "GGLLGG" Output: true Explanation: The robot moves from (0,0) to (0,2), turns 180 degrees, and then returns to (0,0). When repeating these instructions, the robot remains in the circle of radius 2 centered at the origin. Example 2:

Input: "GG" Output: false Explanation: The robot moves north indefinitely. Example 3:

Input: "GL" Output: true Explanation: The robot moves from (0, 0) -> (0, 1) -> (-1, 1) -> (-1, 0) -> (0, 0) -> ...

Note:

1 <= instructions.length <= 100 instructions\[i] is in {'G', 'L', 'R'}

## Solutions

1. **math**
2. there are two cases that the robot would be in a circle
   * after one loop of instructions, the robot's position remain unchanged.
   * the direction are different than the initial direction, we can draw a line between the initial point and the final point after one loop, than as long as robot continues go forward, these lines would eventualy make up a polygan.

```cpp
class Solution {
public:
    int dirs[4][2] = {{0, -1}, {-1, 0}, {0, 1}, {1, 0}};
    void move(const string & s, int & r, int & c, int & curo) {
         for (auto ch : s) {
            if (ch == 'G') {
                r += dirs[curo][0];
                c += dirs[curo][1];
            }
            else if (ch == 'L')
                curo = (curo + 4 - 1) % 4;
            else
                curo = (curo + 1) % 4;
        }        
    }
    bool isRobotBounded(string instructions) {
        int r = 0, c = 0, curo = 1;
        move(instructions, r, c, curo);
        return (r == 0 && c == 0) || curo != 1; 
    }
};
```