Discussion
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 ___.
Starting Arrangement: (H) (H) (H) (T) (T) (T)
Final Arrangement: (T) (T) (T) (H) (H) (H)
Answer: C
The minimum number of steps is 9.
Each step swaps two adjacent coins, so it can change the relative order of only that one pair. In the starting row every one of the 3 heads is to the left of every one of the 3 tails, and in the final row every head is to the right of every tail. So each of the 3 × 3 = 9 head-tail pairs must be reversed, and at least 9 steps are needed.
9 steps are enough: move the leftmost tail three places left (3 steps), then the next tail three places (3 steps), then the last (3 steps). 3 and 6 steps cannot reverse all 9 pairs, and 12 is more than necessary.
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