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 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 below, is ________.
Starting Arrangement: (H) (H) (H) (T) (T) (T)
Final Arrangement: (T) (T) (T) (H) (H) (H)

• A. 3
• B. 6
• C. 9
• D. 12

Hint:

Count how many head-tail pairs must cross past each other to fully reverse the two blocks.

Answer 1

The Correct Option is C

Step 1: Label the coins by type and position.
The starting row is \(H_1, H_2, H_3, T_1, T_2, T_3\) and the target row is \(T_1, T_2, T_3, H_1, H_2, H_3\), so every head must move from the left half to the right half and every tail must move from the right half to the left half.

Step 2: Count how many tails each head must cross.
An adjacent swap only exchanges the positions of one head and one tail sitting next to each other, so each head-tail pair that ends up crossing over needs exactly one swap between that specific head and that specific tail. There are \(3\) heads and \(3\) tails, and in the final order every one of the \(3\) heads must end up to the right of every one of the \(3\) tails.

Step 3: Multiply the crossings.
Each of the \(3\) heads has to cross past each of the \(3\) tails once, giving \(3 \times 3 = 9\) required adjacent swaps in total, and no swap can do the work of two crossings at once.

Final Answer:
The minimum number of steps to fully reverse the two blocks of three coins is nine. \[ \boxed{9} \]