Step 1: Understanding the Concept:
In this sentence, 'on the carpet' suggests the position of the children in relation to the carpet, and 'entranced by the puppet show' implies the effect the puppet show had on the children.
Step 2: Detailed Explanation:
'On' is used to show where the children are sitting, and 'by' is used to show the agent or cause (the puppet show).
Step 3: Final Answer:
Thus, the correct prepositions are "on" and "by".