Question

A coin with heads facing up is shown as H and a coin with tails facing up is shown as T.
Six coins are placed in the Starting Arrangement, as shown in the figure below. A "step" is defined as interchanging a pair of adjacent coins without flipping them. The minimum number of steps needed to go from the Starting Arrangement to the Final Arrangement, as shown in the figure, is ________.

• A. 3
• B. 6
• C. 9
• D. 12

Hint:

Each of the 3 heads must cross past each of the 3 tails at least once, and each adjacent swap fixes exactly one such crossing.

Answer 1

The Correct Option is C

Step 1: Understanding the Question.
Starting row: H H H T T T (positions 1 to 6). Target row: T T T H H H. Each step swaps two coins sitting next to each other. Find the fewest steps needed.

Step 2: Key Formula or Approach.
Track how far the leftmost T coin needs to travel to reach position 1, then the next T to position 2, and so on, since each single-position move of one coin past a neighbor costs exactly one step. Add up the total distance every T coin has to travel to the left.

Step 3: Detailed Explanation.
In the starting row, the three T coins sit at positions 4, 5, and 6. In the target row, T coins must occupy positions 1, 2, and 3.
The T coin at position 4 has to move left to position 1, a distance of $4 - 1 = 3$ steps.
The T coin at position 5 has to move left to position 2, a distance of $5 - 2 = 3$ steps.
The T coin at position 6 has to move left to position 3, a distance of $6 - 3 = 3$ steps.
Adding these up: $3 + 3 + 3 = 9$ steps in total. Every one of these moves is a single adjacent swap with a neighboring H coin, and no swap can move a T coin two places to the left at once, so none of these steps can be skipped or combined.
As a check, the H coins move the same total distance to the right: the H at position 1 moves to position 4 (3 steps), the H at position 2 moves to position 5 (3 steps), and the H at position 3 moves to position 6 (3 steps), also totaling $3+3+3=9$. Both ways of counting agree.

Step 4: Final Answer.
Both the crossing count and the distance count give the same minimum: the row can be rearranged in exactly 9 adjacent swaps, matching option (C). \[ \boxed{9} \]