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: 55 90 61 13 83 14 20
Step I: 55 90 61 13 14 20 83
Step II: 90 61 13 14 20 55 83
Step III: 61 13 14 20 90 55 83
Step IV: 13 14 20 61 90 55 83
Step V: 13 20 14 61 90 55 83
Step VI: 20 13 14 61 90 55 83
Step VI is the last step of this input.
Input: 41 73 29 51 66 36 30
Following the same rule, which number will be third from the left in Step II for this input?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 51
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 right end, and all the other numbers keep their order.
Step I: 83 has digit sum 8 + 3 = 11, the largest among the numbers left, so it moves to the first place from the right
Step II: 55 has digit sum 5 + 5 = 10, the largest among the numbers left, so it moves to the second place from the right
Step III: 90 has digit sum 9 + 0 = 9, the largest among the numbers left, so it moves to the third place from the right
Step IV: 61 has digit sum 6 + 1 = 7, the largest among the numbers left, so it moves to the fourth place from the right
Step V: 14 has digit sum 1 + 4 = 5, the largest among the numbers left, so it moves to the fifth place from the right
Step VI: 13 has digit sum 1 + 3 = 4, the largest among the numbers left, so it moves to the sixth place from the right
After Step VI every one of the numbers is in its final place, so the arrangement is complete.
Apply the same rule to the input 41 73 29 51 66 36 30.
Step I: 66 has digit sum 6 + 6 = 12, the largest among the numbers left, so it moves to the first place from the right, giving 41 73 29 51 36 30 66
Step II: 29 has digit sum 2 + 9 = 11, the largest among the numbers left, so it moves to the second place from the right, giving 41 73 51 36 30 29 66
In Step II, 41 73 51 36 30 29 66, the third from the left is 51.
Why the other options are incorrect
Option (1) 29 — Incorrect. In Step II, 29 is sixth from the left (second from the right).
Option (2) 36 — Incorrect. In Step II, 36 is fourth from the left (fourth from the right).
Option (3) 30 — Incorrect. In Step II, 30 is fifth from the left (third from the right).