Correct Answer: a hard nut to crack

Solution : The correct choice is the third option.

It means a difficult problem or an opponent that is hard to beat, which is appropriate in the context of the sentence. The police are having a difficult time cracking the case.

The meanings of other options are as follows:

• In a nutshell: This phrase is used to summarise or describe something concisely or briefly. It means to present the main points or essence of a topic, situation, or story in a few words.
• Turning over a new leaf: This phrase refers to making a fresh start or changing one's behaviour, habits, or lifestyle for the better.
• In the pink: This idiom is used to describe being in very good health or excellent condition. When someone is "in the pink", it means they are feeling well, healthy, and in a good physical or mental state.

The complete sentence is: "The police have not yet been able to solve the case; it seems to be a hard nut to crack ."