A certain computer program reorders the letters of any seven-letter sequence, and the position of a letter in the new...

1. Translate the problem requirements: The program rearranges letters based on position only. We need to find when repeated applications return us to the original sequence ABCDEFG. The first transformation shows us the pattern: ABCDEFG → DABCGEF.
2. Identify the position mapping pattern: Determine which original position each letter moves to by comparing input ABCDEFG with output DABCGEF.
3. Track the cycle for each position: Follow where each position goes through repeated applications of the transformation until it returns to its starting position.
4. Calculate the overall cycle length: Find the least common multiple of all individual position cycles to determine when all letters return to their original positions simultaneously.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what this computer program actually does. We start with the sequence ABCDEFG and the program transforms it to DABCGEF. The key insight is that the program follows a fixed rule - it always moves letters from specific positions to other specific positions, regardless of what letters are there.

Think of it like a shuffling machine that always moves the card in slot 1 to slot 4, the card in slot 2 to slot 1, etc. The machine doesn't care what cards are there - it just follows the same position-swapping pattern every time.

We need to find how many times we have to run this shuffling process before we get back to our original arrangement ABCDEFG.

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical pattern

2. Identify the position mapping pattern

Now let's figure out exactly where each position goes. We'll compare the original ABCDEFG with the output DABCGEF:

• Position 1 (A) becomes position 2 in output (D_ABCGEF)
• Position 2 (B) becomes position 3 in output (DA_BCGEF)
• Position 3 (C) becomes position 4 in output (DAB_CGEF)
• Position 4 (D) becomes position 1 in output (_DABCGEF)
• Position 5 (E) becomes position 6 in output (DABCG_EF)
• Position 6 (F) becomes position 7 in output (DABCGE_F)
• Position 7 (G) becomes position 5 in output (DABC_GEF)

So our position mapping is: \(1 \rightarrow 2, 2 \rightarrow 3, 3 \rightarrow 4, 4 \rightarrow 1, 5 \rightarrow 6, 6 \rightarrow 7, 7 \rightarrow 5\)

3. Track the cycle for each position

Now let's follow each starting position through multiple runs to see when each returns home:

Starting from position 1:
\(1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1\) (back home after 4 steps)

Starting from position 2:
\(2 \rightarrow 3 \rightarrow 4 \rightarrow 1 \rightarrow 2\) (back home after 4 steps)

Starting from position 3:
\(3 \rightarrow 4 \rightarrow 1 \rightarrow 2 \rightarrow 3\) (back home after 4 steps)

Starting from position 4:
\(4 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow 4\) (back home after 4 steps)

Starting from position 5:
\(5 \rightarrow 6 \rightarrow 7 \rightarrow 5\) (back home after 3 steps)

Starting from position 6:
\(6 \rightarrow 7 \rightarrow 5 \rightarrow 6\) (back home after 3 steps)

Starting from position 7:
\(7 \rightarrow 5 \rightarrow 6 \rightarrow 7\) (back home after 3 steps)

We have two separate cycles: positions 1,2,3,4 form a 4-step cycle, and positions 5,6,7 form a 3-step cycle.

Process Skill: VISUALIZE - Following the concrete movement patterns to see the cycles clearly

4. Calculate the overall cycle length

For all letters to return to their original positions simultaneously, we need both cycles to complete at the same time. This happens at the least common multiple (LCM) of the cycle lengths.

We need: \(\mathrm{LCM}(4, 3)\)

• Multiples of 4: 4, 8, 12, 16, 20, 24...
• Multiples of 3: 3, 6, 9, 12, 15, 18...

The smallest number that appears in both lists is 12.

Therefore, after 12 runs, all positions will have cycled back to their original locations, and we'll have ABCDEFG again.

5. Final Answer

The output will first equal the initial input ABCDEFG after 12 runs.

The answer is C. 12

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the position mapping concept: Students often think the program rearranges letters based on alphabetical order or some other letter-based rule, rather than understanding that it's purely a position-based transformation. They might try to track individual letters (A, B, C, etc.) instead of focusing on how positions (1, 2, 3, etc.) map to each other.

2. Incorrectly identifying the transformation pattern: When comparing ABCDEFG to DABCGEF, students may make errors in determining which original position maps to which new position. This is especially tricky for positions in the middle of the sequence where the pattern isn't immediately obvious.

3. Not recognizing this as a cycle problem: Students might attempt to manually trace through multiple iterations instead of recognizing that this is fundamentally about finding when all positions return to their starting locations simultaneously.

Errors while executing the approach

1. Incorrectly tracing the position cycles: When following each position through multiple iterations (like \(1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1\)), students often lose track or make errors in the sequence, leading to incorrect cycle lengths.

2. Confusing cycle identification: Students may incorrectly group positions into cycles. For example, they might think all 7 positions form one large cycle instead of recognizing the two separate cycles: (1,2,3,4) and (5,6,7).

3. Arithmetic errors in LCM calculation: Even when students correctly identify the cycle lengths as 4 and 3, they may make computational errors when finding \(\mathrm{LCM}(4,3) = 12\), possibly confusing it with other calculations like \(4 \times 3 = 12\).

4. Off-by-one errors in counting: Students might correctly calculate that the LCM is 12 but then second-guess themselves about whether this means 12 runs or 11 runs, especially if they tried to manually verify with a few iterations.