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

Rather than walking word by word, count how many times "add" (the only length-increasing move) can possibly be used among the four moves that take us from word 1 to word 5. Getting from word 1 to word 5 involves exactly 4 moves, and the rule says no operation can repeat immediately after itself.

1. Can all 4 moves be "add"?: no, because "add" would then repeat back to back every single time, which the rules forbid. At most every OTHER move can be "add".
2. So what is the maximum number of "add" moves among 4 moves, with no two adjacent?: out of 4 moves, at most 2 of them can be "add" without any two "add" moves sitting next to each other (for example, add-replace-add-replace, or add-delete-add-replace). Trying to fit 3 "add" moves into 4 slots forces at least two of them to be adjacent, which breaks the rule.
3. What do the remaining 2 moves do?: they must be "replace" or "delete". To keep the length as large as possible, we should use "replace" (no change in length) rather than "delete" (which would shrink the word) for these remaining moves.
4. Add up the effect: starting length is 5. With 2 "add" moves (each +1) and 2 "replace" moves (each +0) spread across the 4 moves from word 1 to word 5, the final length is 5 + 2(11) + 2(00) = 5 + 2 = 7.

Let's summarize:

• Only "add" increases word length, and it can never be used two steps in a row.
• Across 4 moves, at most 2 can be "add" without breaking that rule.
• Using "replace" (not "delete") for the other 2 moves keeps the length from shrinking, giving a maximum final length of 5 + 2 = 7 letters.

So the fifth word can contain at most 7 letters.