Introduction / Context:
This question belongs to the input–output or sequential output tracing category. A device rearranges numbers in several steps following a specific pattern. You must first identify the pattern from a worked-out example and then apply it to a new input sequence to determine at which step the final arrangement is obtained.
Given Data / Assumptions:
- Example input: 85 16 36 04 19 97 63 09.
- Steps for the example:
- Step I: 97 85 16 36 04 19 63 09
- Step II: 97 85 63 16 36 04 19 09
- Step III: 97 85 63 36 16 04 19 09
- Step IV: 97 85 63 36 19 16 04 09
- Step V: 97 85 63 36 19 16 09 04 (final)
- New input: 03 31 43 22 11 09.
- We must find which step is the last when the device processes the new input.
Concept / Approach:
Observe the example carefully to detect the pattern:
- At each step, the device places the largest remaining number from the unsorted part in the next position from the left, keeping the relative order of the other numbers unchanged.
- This is similar to selection sort in descending order where only one element is moved at each step.
Once the numbers are completely in descending order, the device stops and that step is treated as the final step.
Step-by-Step Solution:
Step 1: For the new input 03 31 43 22 11 09, identify the largest number first. The numbers are 3, 31, 43, 22, 11, 9. The largest is 43.
Step 2: Step I: Move 43 to the first position, keeping the others in the original order:
Step I = 43 03 31 22 11 09.
Step 3: Now consider the remaining tail [03, 31, 22, 11, 09]. The largest here is 31.
Step 4: Step II: Place 31 in the second position, keeping the remaining numbers in order:
Step II = 43 31 03 22 11 09.
Step 5: Now the remaining tail is [03, 22, 11, 09]. The largest is 22.
Step 6: Step III: Place 22 in the third position:
Step III = 43 31 22 03 11 09.
Step 7: The remaining tail is now [03, 11, 09]. The largest is 11.
Step 8: Step IV: Place 11 in the fourth position:
Step IV = 43 31 22 11 03 09.
Step 9: The remaining tail is [03, 09]. The largest here is 9.
Step 10: Step V: Place 09 in the fifth position and the remaining number 03 occupies the last position:
Step V = 43 31 22 11 09 03.
Step 11: The sequence is now in complete descending order: 43 > 31 > 22 > 11 > 9 > 3. Therefore, Step V is the last step.
Verification / Alternative check:
We can compare the pattern with the example: in both cases, the device progressively fixes one number at a time in descending order from the left. When there is only one unsorted number left, it automatically occupies the last position and there is no need for an additional step. This confirms that Step V must be the final step for the new input.
Why Other Options Are Wrong:
Step IV: At Step IV, the last two numbers [03, 09] are not in descending order (9 is larger than 3 but comes after it), so the process is not yet complete.
Step VI: There is no need for a sixth step because the arrangement is already fully sorted at Step V.
None of these: This would be correct only if no listed step matched the final arrangement, which is not the case.
Common Pitfalls:
A common error is to assume that the number of steps must always be one less than the number of elements. However, the process can halt earlier if the final two or more elements already fall into the correct order due to earlier moves. Some students also attempt to reorder multiple elements per step, which does not match the given example pattern where only one number is moved on each step.
Final Answer:
For the input 03 31 43 22 11 09, the last step is
Step V.
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