Meaning: 'Dating back' means to have existed since a particular time, in this case, the 17th century, indicating the long-standing nature of the problem.

Wrong options:

• running across: This phrase suggests movement or traversal, which does not fit the context of historical origin.
• taking off: This implies beginning or launching, which is not appropriate for describing the historical timeline of a problem.
• ruling out: This means to eliminate or dismiss, which is contrary to the intended meaning of establishing the problem's historical context.

Correct answer: notoriously

Explanation: The adverb 'notoriously' emphasizes the well-known difficulty of the problem, fitting contextually as it describes the reputation of the three-body problem. It aligns with the sentence's tone and meaning, indicating that the difficulty is widely recognized.

Meaning: 'Notoriously' means being famous or well-known for a particular quality, in this case, the difficulty of solving the problem.

Wrong options:

• closely: This adverb suggests proximity or attention, which does not convey the intended meaning of being widely recognized for difficulty.
• implicitly: This implies something is understood without being stated, which does not fit the context of describing the problem's difficulty.
• arrogantly: This adverb suggests a sense of superiority, which is inappropriate in this context as it does not relate to the problem's difficulty.

Correct answer: extensive

Explanation: The adjective 'extensive' accurately describes the large scale of the computer simulations conducted, fitting grammatically and contextually as it conveys the idea of thoroughness and breadth in the research efforts.

Meaning: 'Extensive' means covering a large area or involving a lot of detail, indicating that the simulations were comprehensive.

Wrong options:

• expanded: This suggests something has been made larger or broader, which does not accurately describe the nature of the simulations.
• expensive: This refers to high cost, which is irrelevant to the description of the simulations' scope or detail.
• exclusive: This implies something is limited or restricted, which contradicts the idea of thoroughness in the simulations.

Correct answer: examined

Explanation: The verb 'examined' fits grammatically and contextually as it indicates that the variables were carefully studied or analyzed, aligning with the scientific nature of the research described.

Meaning: 'Examined' means to inspect or investigate thoroughly, indicating that the researchers analyzed the two-dimensional space of variables in detail.

Wrong options:

• deducted: This means to subtract or take away, which does not fit the context of studying or analyzing variables.
• proposed: This implies suggesting an idea or plan, which does not align with the action of studying the variables.
• spread: This suggests distribution or expansion, which is not relevant to the act of analyzing the variables.

Correct answer: realm

Explanation: The noun 'realm' appropriately denotes a field or domain of study, fitting contextually as it refers to the area of physics being discussed. It aligns with the formal tone of the text.

Meaning: 'Realm' means a domain or area of activity or interest, indicating the specific field of physics related to the theory.

Wrong options:

• survey: This refers to an examination or overview, which does not fit the context of a field of study.
• region: This implies a geographical area, which is not suitable for describing a domain of academic inquiry.
• margin: This suggests an edge or border, which does not relate to the concept of a field or area of study.