Reach a Number

Problem Statement

Imagine you are at the starting point, marked as position 0, on an infinitely long number line. You also have a destination, known as target, located either to the left or right of your starting point. To move towards this target, you can either choose to step to the left or the right. Each move you make is uniquely defined by its order. Specifically, during the ith move, you take exactly i steps. Your objective is to determine the minimum number of moves required to exactly reach your destination.

The problem revolves around finding this minimal sequence of moves, considering at each step you can go in either direction, and the number of steps you take increases linearly with each move. The challenge is to efficiently calculate the minimum moves needed to reach any arbitrary position defined by target.

Examples

Example 1

Input:

target = 2

Output:

3

Explanation:

On the 1st move, we step from 0 to 1 (1 step).
On the 2nd move, we step from 1 to -1 (2 steps).
On the 3rd move, we step from -1 to 2 (3 steps).

Example 2

Input:

target = 3

Output:

2

Explanation:

On the 1st move, we step from 0 to 1 (1 step).
On the 2nd move, we step from 1 to 3 (2 steps).

Constraints

• -109 <= target <= 109
• target != 0

Approach and Intuition

The solution for this problem involves simulating the moves and understanding how the cumulative position changes over progressively increasing moves. Here's how you might approach solving this problem:

1. Understand Movement Dynamics: Every move increases the distance you can cover in i steps by i units. Your aim should always be to minimize numMoves while ensuring the sum of moves covers the target.

2. Manage Directions Optimally:

   • For even numMoves, the sum reaches multiple of numMoves/2.
   • For odd numMoves, the position is centered around (numMoves+1)/2. This understanding helps in figuring out when to move left or right.

3. Calculate Minimum Moves: Start from the first move onward, progressively adding steps (1st move 1 step, 2nd move 2 steps, and so on):

   • Calculate the sum of steps taken as current_total. This is initially 0 and increments as current_total += i, which ensures you move in one optimal direction.
   • If at any point, the absolute difference |current_total - target| can be zeroed out by a correction that is feasible with a number of backward moves (even if spread across multiple steps), then that numMoves is your answer.

Here's an illustration based on examples:

• For target = 2:

  • Move 1 step right: Position = 1 after 1st move.
  • Move 2 steps left: Position = -1 after 2nd move.
  • Move 3 steps right: Position = 2 after 3rd move. Achieving target = 2 in 3 moves.

• For target = 3:

  • Move 1 step right: Position = 1 after 1st move.
  • Move 2 steps right: Position = 3 after 2nd move. Achieving target = 3 in 2 moves.

These considerations will help build an effective algorithm that, through a series of intelligent decision-making processes encompassing direction and magnitude of each move, seeks the minimum moves required to reach the destination.

Solutions

class Solution {
    public int stepsToTarget(int destination) {
        destination = Math.abs(destination);
        int step = 0;
        while (destination > 0)
            destination -= ++step;
        return destination % 2 == 0 ? step : step + 1 + step % 2;
    }
}