DL accepts "on January 2" and rejects "on 2 January". What?
"2 January" is certainly a legit way of writing dates in English. Even BBC uses this: http://news.bbc.co.uk/onthisday/hi/dates/stories/january/2/
I don't think either should be accepted. "2" is not the same as "second/2nd" just like "zwei" isn't the same as "zweite/zweiten". Ordinal numbers are not the same as numerals. You wouldn't say "on January two" or "on two January" to a person would you? It should be "on January 2nd". And if you wanted to put the ordinal number first, you would really say "on the 2nd of January". Leaving out "the" and "of" is just wrong for speech. While it may look OK at the top/bottom of a news article, saying it and using it in sentences is different.
In academic writing it is perfectly acceptable to use only the number before the month. I actually had all the sts nds rds and ths taken out of my PhD by my examiner who insisted that I should write 'the meeting took place on 14 February 1939'. So the BBC looks correct to me.
Probably because there is not a second January in a year. If you wanted to specify the second January in a list of Januaries (or Januarys if you prefer), you would probably include the year as well.
Besides that, I've NEVER heard the preposition on used in conjunction with a month. Something is on a day or in a month.
My birthday is in August. My birthday is on August 21st. My birthday is on the 21st of August.
"On January 2." is extremely vague, especially since the question is about the second and 2nd appears as another correct choice. It's close enough that i might be tempted to choose it, but here you must choose all the correct answers. Is it meant to test whether I know what zweiten means? Well, clearly not since I got it wrong.
Why "On second January" is wrong? You don't have to put "the" for dates in English.
I thought it was "Im" for months and "Am" for days From https://www.duolingo.com/skill/de/Dates-1/tips-and-notes Am, im, um If you want to say "on Monday" and so on, that would be am Montag. Here's a mnemonic to remember when to use which: am Montag um drei Uhr im Juni