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).

The last letter of the English alphabet that the first word of a sentence in the word game can begin with is

• A. t
• B. w
• C. x
• D. y

Hint:

Work backward from "z": a sentence needs enough letters left in the alphabet after the starting letter to place all six words, three per letter at most.

Answer 1

The Correct Option is C

Step 1: Understand what is being asked.
A sentence needs exactly six words, written in increasing alphabetical order, with at most three words allowed to share the same starting letter. We want the LATEST possible starting letter for word 1, so it helps to test the letters near the end of the alphabet first and work backward until one of them actually allows all six words to be completed.

Step 2: Test starting letter "z".
If the first word begins with "z", at most three words in the whole sentence can begin with "z" (the rule's cap). Since "z" is the last letter of the alphabet, there is no letter after it to move on to for the remaining words. So a sentence starting at "z" can hold at most three words in total, not six. Starting letter "z" does not work.

Step 3: Test starting letter "y".
Starting at "y" gives up to three words beginning with "y". After that, the sentence must switch to the next available letter, "z", to keep climbing. But only "z" is left in the alphabet beyond "y", and once the three-word cap on "z" is also reached, there is still no further letter to move into for a sixth word. Working through this shows starting at "y" leaves the sentence one word short of six. Starting letter "y" does not work either.

Step 4: Test starting letter "x".
Starting at "x" gives up to three words beginning with "x". The sentence can then move on to "y", and if needed further to "z", since two full letters ("y" and "z") remain ahead in the alphabet to distribute the remaining three words across, comfortably within the three-per-letter cap. Six words can be completed this way. Starting letter "x" works.

Step 5: Conclude.
Since "z" and "y" both run out of alphabet before six words can be placed, but "x" leaves enough room ahead of it, "x" is the last (latest) letter of the alphabet that the first word of a valid sentence can begin with.

Final Answer:
\[ \boxed{x} \]