Mixing verb tenses involving a Past-Belief/Thought/Attitude about timeless facts

Is the following sentence grammatically correct?

1.) Oscar believed that the sum of two and two is four.

In this case it seems to make the most sense to mix the past tense "believed" with the present-tense "is." If the sentence were changed to,

2.) "Oscar believed that the sum of two and two was four,"

then the meaning of the sentence would be changed.

When the verb is was used, the sentence meant that Oscar had a belief about the present (relative to him). But when the verb was was used, the sentence meant that Oscar had a belief about the past (relative to him).

Consider this sentence that is structurally parallel:

3.) Oscar believed that the killer of J.F.K. was Lee Harvey Oswald.

In this case it seems to make the most sense to not mix verb tenses. Presumably this is because Lee Harvey Oswald's assassination of J.F.K. is an event solely in the past.

I find myself needing to write sentences of this form often, and it always puzzles me whether this mixing of verb tenses is acceptable.

To answer your question, both versions #1 and #2 are acceptable.

It's up to you (or your editor) as to which one you want to use. The second version happens to use a backshifted preterite ("was") in the subordinate clause.

LONG ANSWER: Your question involves the topic of backshifting.

Sometimes backshifting of a subordinate clause is obligatory, sometimes it is optional, sometimes it is not allowed.

Here is an answer to another OP's question that is very similar to yours:

• https://english.stackexchange.com/a/150743/57102

Here are some excerpts of it that are directly related to your question:

There's a common misconception that a present-tense verb being used in its timeless sense (or other related uses) cannot be backshifted. That is untrue, as backshifting is still generally available. For instance, in the older 1985 reference grammar by Quirk et al., A Comprehensive Grammar of the English Language, section 14.31, page 1027:

Here are other examples where present forms may be retained in indirect speech:

• Their teacher had told them that the earth moves around the sun. -- [11]

. . .

In all these sentences, past forms may also be used, by optional application of the backshift rule. Sentence [11] has the simple present in its timeless use, . . .

And so, according to Quirk et al., the following backshifted version (to correspond to [11]) is also acceptable:

• Their teacher had told them that the earth moved around the sun.

Here's a link to a post that I wrote on the general topic of backshifting:

• https://english.stackexchange.com/a/149167/57102

The first sentence is grammatically correct, read this:

On the test Oscar wrote: 2+2=5. But of course, Oscar knew 2 plus 2 is four.

Your choice for 'believed' makes the sentence a bit strange. Here you choose is, because it is stated as a fact.

Generally if you combine two events in the past, you use the same tense. For example:

When the door opened, dad came in.

but there is also the case where event A is happening, when B occurs.

The oven was on when jimmy arrived.

Then there is another possibility, A in the past, B before that:

I played video games after I had done my homework

So there are lots of possibilities, it depends on the context which is correct.