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 the first word in a sentence consists of five letters, then the maximum number of letters that the fifth word in the sentence could contain is

• A. Four
• B. Five
• C. Six
• D. Seven

Hint:

Only the "add" operation increases length, and it can never be used two steps in a row - count how many times it can appear across the four moves from word one to word five.

Answer 1

The Correct Option is D

Step 1: Set up the goal.
The first word has 5 letters. We want the fifth word to have as many letters as possible. Only the "add" operation increases the letter count, and it adds exactly 1 letter each time; "delete" reduces the count by 1, and "replace" keeps the count the same. To maximize length by the fifth word, we should use "add" as often as the rules allow, and avoid "delete" completely.

Step 2: Apply the no-repeat rule to plan the operation sequence.
The rule says no operation can be used twice in a row compared with the word immediately before it. Since we want to avoid "delete" and use "add" as much as possible, the best pattern is to alternate "add" and "replace": add, replace, add, replace, and so on. This way "add" appears every other step, increasing length each time it is used, while "replace" simply fills the steps in between without shrinking anything.

Step 3: Track the length word by word.
Word 1: 5 letters (given).
Word 2: apply "add", length becomes 5 + 1 = 6 letters.
Word 3: apply "replace" (must differ from the previous "add"), length stays at 6 letters.
Word 4: apply "add" (must differ from the previous "replace"), length becomes 6 + 1 = 7 letters.
Word 5: apply "replace" (must differ from the previous "add"), length stays at 7 letters.

Step 4: Check that we cannot do better.
Could word 5 reach 8 letters instead? That would need "add" to be used at word 5 as well, but word 4 already used "add", and the same operation cannot be used twice in a row. So word 5 is forced to use "replace" or "delete", neither of which increases the length. 7 letters is therefore the true maximum for the fifth word.

Final Answer:
The fifth word can contain at most 7 letters. \[ \boxed{7} \]