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: 98 40 44 16 73 92
Step I: 40 98 44 16 73 92
Step II: 40 16 98 44 73 92
Step III: 40 16 44 98 73 92
Step IV: 40 16 44 73 98 92
Step V: 40 16 44 73 92 98
Step V is the last step of this input.
Input: 66 76 41 91 22 17
Following the same rule, which of the following will be Step II for this input?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 22 41 66 76 91 17
Compare each step of the illustration with the one before it. In each step, the number with the smallest 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: 40 has digit sum 4 + 0 = 4, the smallest among the numbers left, so it moves to the first place from the left
Step II: 16 has digit sum 1 + 6 = 7, the smallest among the numbers left, so it moves to the second place from the left
Step III: 44 has digit sum 4 + 4 = 8, the smallest among the numbers left, so it moves to the third place from the left
Step IV: 73 has digit sum 7 + 3 = 10, the smallest among the numbers left, so it moves to the fourth place from the left
Step V: 92 has digit sum 9 + 2 = 11, the smallest among the numbers left, so it moves to the fifth place from the left
After Step V the numbers stand in increasing order of their digit sums, so the arrangement is complete.
Apply the same rule to the input 66 76 41 91 22 17.
Step I: 22 has digit sum 2 + 2 = 4, the smallest among the numbers left, so it moves to the first place from the left, giving 22 66 76 41 91 17
Step II: 41 has digit sum 4 + 1 = 5, the smallest among the numbers left, so it moves to the second place from the left, giving 22 41 66 76 91 17
So Step II is 22 41 66 76 91 17.
Why the other options are incorrect
Option (2) 22 66 76 41 91 17 — Incorrect. This is Step I; in Step II 41 must also move.
Option (3) 22 41 76 66 91 17 — Incorrect. Here 41 changes places with 66; the machine moves 41 up and the other numbers keep their order.
Option (4) 22 41 17 66 76 91 — Incorrect. This is Step III, where 17 has also been moved.