It is necessary that children be honest all the time. (Is this sentence correct?)

It is necessary for children to be honest all the time.

They're both correct and have the same meaning. The first one is a bit higher register.
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Thanks a lot! Can you insert "should" in the first sentence? ...should be honest all the time.

And how do you pronounce "the" in "the same"? Is it "thuh or thee"?

Thanks again!!!
I'm going to pass on the "should" question. My instinct tells me it could be ambiguous, because "should" has several uses. If you use "should" in the sense of "ought to," then it would be redundant. You'd be saying it's necessary that they ought to do it. And, in fact, you'd no longer be saying that they should do it. You'd be making a major change in the meaning. I hope I'm making this clear. We're talking about two things, the obligation to be honest, and the act of being honest. If you insert "should," you're now saying that the obligation is necessary, and you're no longer saying that the act is necessary.

But I have this feeling there's another sense of "should" which would be permissible - like "if it should happen to rain." (Maybe I'm starting to lose that feeling.)

At this point, I have to advise against it. You may shorten your sentence to "Children should be honest all the time," and it would mean about the same thing, although I believe "necessary" suggests a stronger obligation than "should," perhaps approaching compulsion.

On the pronunciation question, it's "thuh same." We use "thee" only before (non-silent) vowels, like "thee apple," or when the vowel is preceded by a silent consonant, as in "thee honest man."
I got it! It's clear now. Thanks Avangi!!!
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