A number arrangement machine, when given an input line of numbers, rearranges them step by step following a particular rule. An illustration of an input and its rearrangement is given below.
Input: 22 48 37 85 74 69 97
Step I: 97 22 48 37 85 74 69
Step II: 97 69 22 48 37 85 74
Step III: 97 69 85 22 48 37 74
Step IV: 97 69 85 48 22 37 74
Step V: 97 69 85 48 74 22 37
Step VI: 97 69 85 48 74 37 22
Step VI is the last step of this input.
Step II of another input is: 88 58 51 34 80 37 56
Following the same rule, which of the following will be Step IV for that input?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 88 58 56 37 51 34 80
Compare each step of the illustration with the one before it. In each step, the number with the largest digit sum (the sum of its two digits), among the numbers not yet arranged, moves to the next place from the left end, and all the other numbers keep their order.
Step I: 97 has digit sum 9 + 7 = 16, the largest among the numbers left, so it moves to the first place from the left
Step II: 69 has digit sum 6 + 9 = 15, the largest among the numbers left, so it moves to the second place from the left
Step III: 85 has digit sum 8 + 5 = 13, the largest among the numbers left, so it moves to the third place from the left
Step IV: 48 has digit sum 4 + 8 = 12, the largest among the numbers left, so it moves to the fourth place from the left
Step V: 74 has digit sum 7 + 4 = 11, the largest among the numbers left, so it moves to the fifth place from the left
Step VI: 37 has digit sum 3 + 7 = 10, the largest among the numbers left, so it moves to the sixth place from the left
After Step VI the numbers stand in decreasing order of their digit sums, so the arrangement is complete.
In Step II the first two numbers, 88 58, are already in place. Continue from Step II with the same rule.
Step III: 56 has digit sum 5 + 6 = 11, the largest among the numbers left, so it moves to the third place from the left, giving 88 58 56 51 34 80 37
Step IV: 37 has digit sum 3 + 7 = 10, the largest among the numbers left, so it moves to the fourth place from the left, giving 88 58 56 37 51 34 80
So Step IV is 88 58 56 37 51 34 80.
Why the other options are incorrect
Option (1) 88 58 56 37 34 80 51 — Incorrect. Here 37 changes places with 51; the machine moves 37 up and the other numbers keep their order.
Option (3) 88 58 56 51 34 80 37 — Incorrect. This is Step III; in Step IV 37 must also move.
Option (4) 88 58 56 37 80 51 34 — Incorrect. This is Step V, where 80 has also been moved.