The classic fantasy novel The Hobbit tells of a quest by the hobbit Bilbo Baggins and 13 dwarves with the aid of Gandalf the wizard to take back the kingdom under Lonely Mountain from Smaug the fearsome dragon. Since Smaug would confront them if they tried to go in the front way, the company's only hope is to find a secret door into the mountain.

At Bilbo's home before the journey begins, Gandalf takes out a map that had belonged to Thror, the grandfather of Thorin, the dwarf who is leading the quest. It is a plan of the Lonely Mountain with a rune showing a spot on the mountain's west side that marks the location of the secret door. Gandalf also has a key that opens the door once they find it.

In Rivendell, the home of the elves, Elrond reveals more secrets of the map. He says that it has writing in moon-letters, which are runes that can only be seen when the moon shines behind them, and in this case particularly, it refers to a crescent moon on a midsummer's eve. Elrond reads:

Stand by the grey stone when the thrush knocks and the setting sun with the last light of Durin's Day will shine upon the key-hole.

Thorin then explains that Durin's Day is the first day of the dwarves' New Year, which falls on the first day of the last moon of autumn on the threshold of winter.

When Bilbo and the dwarves find the shut secret entrance on the western slope of Lonely Mountain, at first they try to force it open but are unsuccessful. Only at the proper moment on Durin's Day does the key-hole reveal itself, and then they are able to open the door and gain access to the interior of the mountain.

When the dwarves and Bilbo find themselves on the Lonely Mountain at the secret door, they are reminded that the secret entrance is revealed by the last light of the setting sun on Durin's Day, according to the map they had. As it happens, they are there on Durin's Day, and the light of the sun as it sets reveals a keyhole.

One way to know this is that it describes Durin's Day in the book itself. Durin's Day can only occur on the last new moon of Autumn, and only then when the sun and moon are in the sky at the same time. It is thus a somewhat rare date. In our calendar system it would occur somewhere between November 22 and December 21. Tolkien's calendar system works a bit differently, and the likely date for Durin's day is pushed up to between October 23 and November 22.

In The Hobbit, on what special day do Bilbo and the dwarves discover a way into the Lonely Mountain and how do you know?

In addition to what the previous educator has noted before me, I would note that The Hobbit does contain a description when its protagonists reach the mountain, and this does match up fairly well with the timeframe provided in chapter 3. In chapter 11, where they reach the mountain, Tolkien provides the following passage:

Only in June they had been guests in the fair house of Elrond, and though autumn was now crawling towards winter that pleasant time now seemed years ago. (203-204)

From here, they spend some unknown span of time trying to locate the door. Later, after they have actually found it and shortly before Bilbo manages to get them entry within, Tolkien provides the following conversation, giving us an even more precise measurement of time:

"Tomorrow begins the last week of Autumn," said Thorin one day.

"And winter comes after autumn," said Bifur.

"And next year after that," said Dwalin. (207)

Like the previous educator, I cannot say that this connects perfectly with Durin's Day. Looking back towards Durin's words from chapter 3, things are somewhat unclear and I think open to a certain degree of interpretation, given that he defines Durin's Day in terms that are more poetical than precise (it's all about the sun and moon), and this lends a certain ambiguity to his words. Generally speaking however, we're looking at the same time-frame regardless: the very end of Autumn and the beginning of the new year. The connection is there.

In The Hobbit, on what special day do Bilbo and the dwarves discover a way into the Lonely Mountain and how do you know?

We are given the answer to this question at the end of Chapter 3, when Elrond helps them work out where they need to go and tells them about moon runes. Reference is made to "Durin's Day," which pleases the dwarves incredibly because Durin is actually an ancestor of Thorin and a very important person for dwarves in general. Note what is said about Durin's day:

"The first day of the dwarves' New Year," said Thorin, "is as all should know the first day of the last moon of Autumn on the threshold of Winter. We still call it Durin's Day when the last moon of Autumn and the sun are in the sky together."

So, reference is made to the importance of Durin's day, however it is not clear in the novel if they actually discover the way in to the mountain on Durin's Day itself - no mention in Chapter 4 is made of any particular day except for the fact that winter is coming and it is getting much colder.