Final answer:
Suspect intussusception in a child with episodes of intense vomiting and abdominal pain interspersed with periods of normalcy.
Step-by-step explanation:
When a child presents with symptoms of vomiting and abdominal pain that occur in sudden, intense bouts followed by periods of seeming wellness, one should suspect intussusception. This condition is characterized by a part of the intestine sliding into an adjacent part of the intestine, leading to such intermittent pain.
In contrast, appendicitis usually presents with continuous and progressively worsening pain, gastroenteritis often includes diarrhea and fever, and food allergies can trigger a range of reactions, including hives, swelling, and in some cases gastrointestinal discomfort.