A coin with heads facing up is shown as a circled letter "H", and a coin with tails facing up is shown as a circled letter "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 ________.
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Every step can only swap two coins that sit next to each other. Count how many H, T pairs are in the wrong relative order in the Starting Arrangement; each needs exactly one swap to fix.
Step 1: Write down the starting and final arrangements as sequences. Number the six positions 1 through 6. The Starting Arrangement is \[ H,\ H,\ H,\ T,\ T,\ T \] and the Final Arrangement is \[ T,\ T,\ T,\ H,\ H,\ H \]
Step 2: Understand what one step can do. A step swaps two coins that are sitting right next to each other, without flipping either coin. So each step can move a coin only one position to the left or right at a time.
Step 3: Count how far each coin must travel. In the Final Arrangement, all 3 T coins must end up to the left of all 3 H coins. Right now, every H coin sits to the left of every T coin. So every single H, T pair that is currently in the wrong relative order (H before T) must be swapped past each other exactly once to fix the order.
Step 4: Count the number of such out of order pairs. There are 3 H coins and 3 T coins, and in the Starting Arrangement every H is before every T. So the number of H, T pairs that are in the wrong order is \[ 3\times3=9 \] Each of these 9 pairs needs exactly one adjacent swap between an H and a T to bring the sequence to the Final Arrangement, and swapping two coins of the same type (H with H, or T with T) is never needed since same type coins are already indistinguishable in the final pattern.
Step 5: Confirm this is the minimum. Every adjacent swap of two different coins (one H and one T) can fix at most one out of order pair. Since there are exactly 9 such pairs to fix, at least 9 steps are required, and the count above shows 9 steps are also sufficient. So 9 is both necessary and enough.
Step 6: Rule out the other options.
(A) 3: Far too few. Even moving just the front H coin past all 3 T coins alone would already take 3 steps, and there are two more H coins that also need to cross all the T coins. Incorrect.
(B) 6: This would be the count if only 2 H coins needed to cross the 3 T coins (\(2\times3=6\)), but there are 3 H coins to move, not 2. Incorrect.
(D) 12: This overcounts; it is more than the total number of out of order H, T pairs, so it is more swaps than necessary. Incorrect.
Final Answer: The minimum number of steps needed is \[ \boxed{9} \]
Find dV/dt