Directions : A word and number arrangement machine when given an input line of…

Concept

In a word-and-number arrangement machine, every step applies one fixed rule to each word and each number. The job is to decode those per-step rules from the worked illustration and then replay exactly the same rules on the new input line. Words and numbers follow separate rules, and a number keeps its own position while the words are re-sorted.

Rules decoded from the illustration

Words:

• Order: vowel-initial words first in alphabetical order, then consonant-initial words in alphabetical order.

• Step I → II (cipher): every vowel shifts to the next letter, every consonant to the previous letter.

• Step II → III: delete all vowels, then delete every letter that appears more than once.

Numbers:

• Input → Step I: replace the first digit by its square, add 1 to each middle digit, subtract 1 from the last digit.

• Step I → II: rearrange the digits — even digits in ascending order, then odd digits in ascending order.

• Step II → III: split the digit string at the first fall (the first digit smaller than the one before it; if it only rises, split at the middle); each Step III number is the digit-sum of one part.

• Step III → IV: taking the left value a and the right value b, the Step IV number = b² − a.

• Step IV → V: add 3 when a+b is odd, add 4 when a+b is even.

Applying the rules to the new input

Input: 8548 UNITY 4369 FRAME 3657 EVENT 5378 SPOIL 6878 PLANK.

Words become:

• Step I order: EVENT, UNITY, FRAME, PLANK, SPOIL

• Step II: FUFMS, VMJSX, EQBLF, OKBMJ, ROPJK

• Step III: MS, VMJSX, QBLF, KBMJ, RPJK

Number columns (each number keeps its place):

Evaluating the statements

1. Statement I: In Step III the words are MS, VMJSX, QBLF, KBMJ, RPJK. Three of them — VMJSX, QBLF, RPJK — begin with a letter that comes after M (V, Q, R). Exactly three words qualify, so Statement I is TRUE.

2. Statement II: The Step V numbers are 132, 50, 248, 134, 350 — every one is even, so there are zero odd numbers. “More than two odd numbers” is not met, so Statement II is FALSE.

3. Statement III: The Step IV numbers are 128, 46, 244, 130, 347 — four of them (128, 46, 244, 130) are even, which is more than two, so Statement III is TRUE.

Cross-check

A number’s Step IV value is b² − a, whose parity equals the parity of a + b, and a + b is simply the digit-sum of that Step I number. Four of the five Step I numbers have an even digit-sum, matching the four even Step IV values. Each Step V number adds 3 to an odd Step IV value or 4 to an even one — an odd+odd or an even+even sum in either case — so every Step V value is even, confirming there are no odd numbers in Step V.

So Statements I and III are correct and II is not; the answer is “Both I and III.”