The correct Active/Passive form of the given sentence is:

☀ He will have to be amused with riddles and bed-time stories.

Explanation for other options:

☀ He had to be amused with riddles and bed times stories. This option changes the tense from future to past ("had to be amused") and does not represent the same meaning as the original sentence.

☀ He had been amused by riddles during bed times stories. This option changes the meaning of the sentence by using "had been amused" instead of "will have to be amused." The original sentence states a future obligation, not a past event.

☀ He will have to be amused with riddles and bed times stories. This option has the correct passive form, using "will have to be amused" to indicate the future obligation to amuse "Him" with riddles and bed-time stories. This option is grammatically correct and maintains the original meaning of the sentence.