Question

In a game, a "word" is any combination of at least five letters of the English alphabet (the word does not have to have a meaning). A "sentence" in this game consists of exactly six such words, and it must satisfy all of the following rules:
1. The six words are written from left to right on a single line, in strictly increasing alphabetical order.
2. The sentence may start with any word. Each successive word is obtained from the word right before it by applying exactly ONE of three operations: deleting one letter, adding one letter, or replacing one letter with another letter.
3. At most three of the six words can begin with the same letter.
4. Apart from the first word, every word must be formed using an operation different from the one used to form the word immediately before it (so the same operation can never be used twice in a row).

If "clean" is the first word in a sentence and "learn" is another word in the sentence, then which one of the following is a complete and accurate list of the positions "learn" could occupy?

• A. Third
• B. Second, third, fourth
• C. Third, fourth
• D. Third, fourth, fifth, sixth

Hint:

First find the fewest moves needed to turn "clean" into "learn" - that fixes the earliest possible position; later positions come from inserting extra detour words.

Answer 1

The Correct Option is D

Step 1: Find the earliest position "learn" can occupy.
The first word is "clean" (5 letters: c-l-e-a-n). To reach "learn" (5 letters: l-e-a-r-n), compare the two words. Removing the leading "c" from "clean" gives "lean" (4 letters: l-e-a-n), and then adding an "r" before the final "n" gives "learn" (l-e-a-r-n). That is two moves: delete "c", then add "r". Since two separate moves are needed, "learn" cannot appear as the second word (only one move away from "clean"); the earliest it can appear is the THIRD word, after "clean" then "lean" then "learn". So any option built around "second" as a possible position is automatically wrong.

Step 2: Confirm "learn" as the third word.
Chain: clean (delete "c") lean (add "r") learn. Two operations, delete then add, different from each other, satisfying the no-repeat rule. Alphabetically, "clean" comes before "lean", which comes before "learn", so the order climbs correctly. Position THREE is valid.

Step 3: Push "learn" to the fourth position.
Insert one extra word before reaching "learn" while still using a fresh operation each step. Chain: clean (replace "n" with "r") clear (delete "c") lear (add "n") learn. That is replace, delete, add: three different consecutive operations, and "clean", "clear", "lear", "learn" climb alphabetically in that order. Position FOUR is valid.

Step 4: Push "learn" to the fifth position.
Insert two extra words before "learn". Chain: clean (replace "c" with "d") dlean (delete "d") lean (replace "n" with "r") lear (add "n") learn. Operations used: replace, delete, replace, add, each different from the one right before it, and the words climb alphabetically: clean, dlean, lean, lear, learn. Position FIVE is valid.

Step 5: Push "learn" to the sixth (last) position.
Insert three extra words before "learn". Chain: clean (replace "c" with "d") dlean (add a letter, for example "p") dleapn (replace "d" with "e") eleapn (delete "p") leapn (replace "p" with "r") learn. Operations: replace, add, replace, delete, replace, and each is different from its immediate neighbour, while the six words climb alphabetically at every step. Position SIX is valid, meaning "learn" can be the sixth and final word too.

Final Answer:
"Learn" can occupy the third, fourth, fifth, or sixth position, but never the first or second. \[ \boxed{\text{Third, fourth, fifth, sixth}} \]