How to Solve Moving Stones Until Consecutive

In this problem, you're given three stones on a number line at different positions, and the goal is to move them into three consecutive positions. The task is to determine the minimum and maximum number of moves required to achieve this, based on the constraints of the game.

Problem Statement

You are given three stones at distinct positions on the X-axis, represented by integers a, b, and c. In each move, you pick up a stone at an endpoint and place it in an unoccupied position between the remaining two stones, making sure the stone does not overlap with the other stones. The game ends when all three stones are at consecutive positions on the number line.

The task is to determine the minimum and maximum number of moves required to arrange the stones in three consecutive positions. You need to analyze how to make optimal moves that minimize the number of steps while also considering possible maximum moves, given the flexibility in choosing which stone to move.

Examples

Example 1

Input: a = 1, b = 2, c = 5

Output: [1,2]

Move the stone from 5 to 3, or move the stone from 5 to 4 to 3.

Example 2

Input: a = 4, b = 3, c = 2

Output: [0,0]

We cannot make any moves.

Example 3

Input: a = 3, b = 5, c = 1

Output: [1,2]

Move the stone from 1 to 4; or move the stone from 1 to 2 to 4.

Constraints

• 1 <= a, b, c <= 100
• a, b, and c have different values.

Solution Approach

Minimum Moves

To determine the minimum number of moves, the strategy is to move one stone adjacent to another, then position the third stone. This can always be done in at most two moves. The key observation is that if the stones are already close to each other, fewer moves are required, and it's possible to make the stones consecutive with one or two steps.

Maximum Moves

The maximum number of moves occurs when the stones are positioned far apart. In the worst case, the maximum number of moves is three, which involves moving the stones one by one to place them in consecutive positions. The challenge is understanding how to move the stones to create a valid set of three consecutive positions without skipping any.

Optimal Move Selection

A key insight into solving this problem is selecting which stone to move for each step. Whether you pick the leftmost or rightmost stone affects how quickly you can make the stones consecutive. Always prioritize moving the stone that allows you to create consecutive positions in the fewest moves.

Complexity Analysis

The problem can be solved with constant time and space complexity, as the positions of the stones are limited to a fixed number of possibilities. The main challenge lies in calculating the minimum and maximum number of moves based on the initial positions of the stones, which can be done in constant time.

What Interviewers Usually Probe

• The candidate demonstrates understanding of how to minimize the number of moves in the game.
• The candidate efficiently identifies the minimum and maximum number of moves required for the given input.
• The candidate discusses the strategy for choosing which stone to move in each step.

Common Pitfalls or Variants

Common pitfalls

• Overcomplicating the calculation of moves when the solution is simple and requires minimal operations.
• Failing to recognize that the minimum moves are always limited to a maximum of two steps.
• Misunderstanding the maximum number of moves and assuming that more than three steps might be required.

Follow-up variants

• Consider the case where the stones are placed in consecutive positions from the beginning.
• Explore how the problem changes if the stones can be moved to any integer position instead of being restricted to unoccupied spaces between the other stones.
• Examine how the problem would change if there were more than three stones, requiring a strategy for larger sets of stones.

How GhostInterview Helps

• GhostInterview helps by providing a clear understanding of how to approach the problem's patterns, especially the difference between minimum and maximum move calculations.
• It guides the user through optimal strategies for solving the problem, focusing on the most efficient approach.
• GhostInterview assists with identifying common mistakes, such as miscalculating the minimum number of moves or overcomplicating the solution.

Topic Pages

• Math
• Brainteaser

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FAQ

What is the minimum number of moves required in "Moving Stones Until Consecutive"?

The minimum number of moves required is always at most two, by placing one stone adjacent to another, then moving the third stone next to the others.

What is the maximum number of moves in this problem?

The maximum number of moves is three, as you can move each stone one by one to make them consecutive.

How can GhostInterview help with "Moving Stones Until Consecutive"?

GhostInterview helps by offering detailed insights into how to approach the problem, highlighting optimal strategies, and addressing potential mistakes.

Can the stones start already in consecutive positions?

Yes, if the stones are already consecutive, no moves are required, and the answer would be [0,0].

How do I calculate the number of moves in "Moving Stones Until Consecutive"?

To calculate the number of moves, analyze the distance between the stones and determine how many steps are needed to place the stones consecutively, based on the game rules.