The English versions are correct in translating "on the third day."
To take Matthew 16.21 as an example, this is the relevant section in the Greek.
καὶ τῇ τρίτῃ ἡμέρᾳ ἐγερθῆναι
[Jesus began to explain that] on the third day he would be raised to life.
The key to translating this clause is to understand the declension or form of the words. In English the general principle is that words have a single grammatical form, and meaning is determined by the word order. Thus "The man loved the woman" has a different meaning from "The woman loved the man." But in Greek the words themselves have different forms which determine the meaning. If a Greek writer wanted to say that the man loved the woman, he would give the word "man" a specific ending to indicate that. Word order would then be irrelevant.
(An analogy might be if in English we marked the subject of a sentence by putting that word in capitals. "The MAN loved the woman." Then "MAN woman loved" and "woman MAN loved" would mean the same thing.)
Now in NT Greek there were four main forms of a noun:
In the above clause the words τῇ τρίτῃ ἡμέρᾳ are in the dative form. In general terms this form marks the indirect object of an action.
"John bought the present for his wife."
"John bought the present at the mall."
"John bought the present at 4 pm."
In each case John is the one who did the buying (nominative) and the present is the thing bought (accusative). The remaining words will be in the dative case because they say something extra about the action: why, where or when it happened. They relate to but are not the same as the object of the action.
In Matthew 16.21 Jesus says only three words in the Greek. But to bring out the form of those words in English we need to add another word. The Greek words are dative and in context are describing WHEN the action will take place. The third day is the point at which there will be a resurrection, says Jesus. The only way we can read that in English is by saying "on the third day."
We see the same process in other passages. In John 19.2 we read that the soldiers put a crown of thorns on Jesus head (ἐπέθηκαν αὐτοῦ τῇ κεφαλῇ). The object is the crown of thorns; "his head" is in the dative form and in context tells us where they put it. So we need to add "on" to get the same sense - "on his head". In the same way Mark 16.2 records the women going to the tomb "(on) the first day of the week." The dative form tells us this is when they went.
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