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

This puzzle is really about counting inversions: an adjacent swap can fix exactly one out-of-order pair, so the minimum number of steps equals the number of pairs that are in the wrong relative order between the start and the end.

1. Option (A), 3: three swaps could only fix three misplaced pairs, but far more pairs are out of order here.
2. Option (B), 6: six swaps still fall short, since every one of the $3$ heads is out of order with respect to every one of the $3$ tails.
3. Option (C), 9: each of the $3$ heads needs to cross each of the $3$ tails exactly once to move from the left block to the right block, giving $3 \times 3 = 9$ out-of-order pairs, and hence $9$ adjacent swaps.
4. Option (D), 12: this overcounts the crossings; coins of the same face never need to swap past each other, so $12$ is more moves than necessary.

Counting every head-tail crossing gives exactly $9$ required swaps, so option (C) is correct.