After retreating in his room and hearing the words of Gania, the Prince asks Kolya to lead him to his father:

I wished to ask you whether you could show me the way to Nastasia Philipovna’s tonight. I must go; I have business with her; I was not invited but I was introduced. Anyhow I am ready to trespass the laws of propriety if only I can get in somehow or other.

However, the Prince could not have chosen a worse guide: the general Ivolgin is tipsy, and keeps confusing addresses and people and insists on entering houses, knocking on doors and introducing the Prince to people who are not who he claims them to be. Eventually, however, the Prince is saved by Kolya who finds them at one of those houses and immediately proposes to show him the right way:

Nastasia Philipovna? She does not live there, and to tell you the truth my father has never been to her house! It is strange that you should have depended on him! She lives near Wladimir Street, at the Five Corners, and it is quite close by. Will you go directly? It is just half-past nine. I will show you the way with pleasure.

Kolya leads the Prince to the place, a house at a crossroad called “The Five Corners”. In effect, the whole chapter was nothing but an excuse to lose time for the Prince. However, Dostoyevsky can’t help this time to leave the philosophising to the shrewd Kolya:

Well, whether you go on business or not is your affair, I do not want to know. The only important thing, in my eyes, is that you should not be going there simply for the pleasure of spending your evening in such company—cocottes, generals, usurers! If that were the case I should despise and laugh at you. There are terribly few honest people here, and hardly any whom one can respect, although people put on airs—Varia especially! Have you noticed, prince, how many adventurers there are nowadays?

Once again the children are the more intelligent and meaningul people in that crazy city!

When the Prince arrives at Nastasia Filippovna’s home, he finds there Gania again, but also Totski and the general Epanchin, along with several new people - and all are sort of awaiting Nastasia’s big decision about the wedding. But it feels more like a funeral than a real party:

The rest of the guests [...] not only had no gift for enlivening the proceedings, but hardly knew what to say for themselves when addressed. Under these circumstances the arrival of the prince came almost as a godsend.

However, once again the revelations are delayed and someone (Ferdischenko) has a great idea of a game:

each of us, without leaving his place at the table, should relate something about himself. It had to be something that he really and honestly considered the very worst action he had ever committed in his life. But he was to be honest—that was the chief point! He wasn’t to be allowed to lie.”

The idea is welcomed with surprise, nervosity, disgust, and doubt - but also with a great deal of fascination and Nastasia seizes this opportunity to gain time. And on that cliffhanger, the chapter XIII ends...

Before to close that article, I would like however to remind something: after more than ten chapters, we are still in the same day of the end of November. We remember that the first chapter was in a train, and it was roughly 9 o'clock in the morning. That means Myshking arrived somewhere around 10 o'clock in Saint-Petersburg.

He arrived at a train station which is not named, but it makes sense to assume that it is the former Warsaw Station which now houses the Central Museum of Railway Transport. Therefore, I have reconstructed from this point on his journey:

First, he accompanies Rogojin, who is going to Vorozensky… Therefore it’s a straight line from the station.

Arriving at the Fontanka river embankment, they separate and the Prince hires a coach which probably follows all

the way the Fontanka river

Now, Prince arrives at Epantchin (General Epanchin lived in his own house near the Litaynaya.) where he even stays

for lunch!

Then, walking, he goes to Gania’s house (unknown address), where a kerfuffle erupts, but he has no time to stay because… He wants to goes to Nastasia’s house and wanders the streets with the general Ivolgin…

And is finally led by Kolya to the house of Nastasia Filippovna who lives at the Five Corners: "She lives near Wladimir Street, at the Five Corners, and it is quite close by. Will you go directly? It is just half-past nine."

So, here is my best guess about the whole trajectory of the Prince:

In total, the Prince Myshkin has so far, in his first day in Saint-Petersburg, covered almost 8 kilometers. Not bad at all!

These two chapters don't really make the plot advance, they just make Myshkin walk a lot. First with the general Ivolgin and then with Kolya, in search for the apartments of Nastasia Filippovna. Something is going on there: it is supposed to be the place and time for the resolution of that fateful wedding! But before we can learn her decision, they all have to play a game called "who is the most despicable human being" - sort of.