We traditionally observe the fast commemorating the destructions of the First and Second Temples on the 9th of the month of Av. Why did the Sages establish the fast on this particular date?  Surprisingly, this date is not mentioned in the Tanakh as the day the Temple was destroyed.  Instead, the Tanakh presents us with two conflicting options.

II Kings 25: 8-9 states: "Now in the fifth month, on the seventh day of the month, which was the nineteenth year of king Nebuchadnezzar, king of Babylon, came Nebuzaradan the captain of the guard, a servant of the king of Babylon, unto Jerusalem... And he burnt the house of the LORD, and the king's house; and all the houses of Jerusalem, even every great man's house, burnt he with fire."

According to the source above, Tisha B'Av should be observed on the 7th of Av, because that was the date of the destruction of the Temple.

However, Jeremiah 52: 12-13 presents us with a conflicting date: "Now in the fifth month, in the tenth day of the month, which was the nineteenth year of king Nebuchadrezzar, king of Babylon, came Nebuzaradan the captain of the guard, who stood before the king of Babylon, into Jerusalem... and he burned the house of the LORD, and the king's house; and all the houses of Jerusalem, even every great man's house, burned he with fire."

This source suggests that Tisha B'Av should be observed on the 10th of Av. Why are there two different dates?  Even more surprisingly, as stated above, we do not observe Tisha B'Av on either one of these dates!

The Talmud in Tractate Ta'anit 29a addresses this question and suggests a resolution:

"How [do we resolve this contradiction]? 

On the seventh, gentiles entered the Sanctuary, and they ate and befouled it on the seventh, eighth, and ninth. close to nightfall [on the ninth], they set upon it the fire, and it burnt continuously all the entire day, as it is said, 'Woe to us, for the day has gone, for the shadows of the evening are stretched out (Jeremiah 6:4).' And this is [the reasoning] that Rabbi Jochanan said, 'If I had been in that generation, I would not have established it [on the ninth] but rather on the tenth, because most of the Sanctuary was burnt on it.' And the Rabbis [reasoning was that] the earliness of suffering was preferred."

The Rabbis decreed that Tisha B'Av should be observed on the 9th of Av because that is when the fire actually began to burn. However, many of the mourning customs that apply to the Nine Days and Tisha B'Av carry onto the 10th day as well, because it was on that day that the majority of the Temple burned.

May we merit to see these days turn from ones of fasting and mourning to happiness and celebration.