Moving Stones Until Consecutive

Problem Description

There are three stones in different positions on the X-axis. You are given three integers a, b, and c, the positions of the stones. In one move, you pick up a stone at an endpoint (i.e., either the lowest or highest position stone), and move it to an unoccupied position between those endpoints. The game ends when you cannot make any more moves (i.e., the stones are in three consecutive positions). Return an integer array answer of length 2 where answer[0] is the minimum number of moves you can play, and answer[1] is the maximum number of moves you can play.

Key Insights

• The stones can be represented as three integers on a number line.
• The goal is to make the stones occupy three consecutive positions.
• Moving the stone at the highest or lowest position provides opportunities to reduce the distance between stones.
• The minimum and maximum moves depend on the gaps between the stones.

Space and Time Complexity

Time Complexity: O(1)
Space Complexity: O(1)

Solution

To solve the problem, we can follow these steps:

1. Sort the three positions to identify the lowest (x), middle (y), and highest (z) positions.
2. Calculate the gaps between the stones:
   • gap1 = y - x (distance between the first and second stones)
   • gap2 = z - y (distance between the second and third stones)
3. Determine the minimum moves:
   • If both gaps are 1, the stones are already consecutive (0 moves).
   • If one gap is 1 and the other is greater than 1, only one move is needed.
   • If both gaps are greater than 1, two moves are needed.
4. Calculate the maximum moves:
   • If the gaps are both greater than 1, the maximum moves can be calculated as the number of gaps - 1. If one gap is 1, then it’s the gap size of the other - 1.

This approach uses basic arithmetic and condition checks to derive the result efficiently.

Code Solutions