May the Best Thief Win - Amantius Allectus
Regardless of how you learn of the Thieves Guild entrance exam, head over to the Waterfront district and enter the Garden of Dareloth at around midnight (wait if you have to do so). It’s just a little corner past a few beggars’ bedrolls behind an abandoned house. Armand Christophe is the Imperial City Doyen and the man that we have to impress to enter the Thieves Guild. He also holds a bright torch, so he’s easy to find. You may have to boost his disposition a bit to get him to tell you about the test, although mentioning the beggar’s approval (through the “Gray Fox" option) should help.
Regardless, he’ll explain the unique test for the unique circumstance. Since there are three potential recruits, he’s requiring a competition. This is a little tricky.
Standard theft rules apply. You can’t kill anyone involved in the quest while it’s active. Other than that, it’s pretty much your call. Remember that you aren’t in the Thieves Guild yet, so their more strict rules don’t apply.
The diary belongs to Amantius Allectus. He’s an alchemist who lives in the Temple District of Imperial City. You can get a marker for his house by asking any beggar about him and paying 5 coins. There's one sleeping on the bedroom by the wall right outside the Garden of Dareloth.
Note that, yes, Amusei is an idiot. He’ll just wander around the Elven Gardens and the rest of the city aimlessly while looking for the house. He’ll still be at it the next day.
Methredhel, on the other hand, is a genius thief. She already knows where Amantius Allectus lives and she’ll sprint to his house to grab the diary. It may not actually be possible to beat her. If you’re not built as a thief, then don’t worry. We have several other options. If you want to try, then you can try to outrun her. It will be hard for a new character though since she is really fast.
Fast travel can even the game, since I believe you’ll both move at the “default" speed and end up in the Temple District at the same time. You may be able to cut a corner and beat Methredhel there. If she makes it there before you, then you might still be able to beat her. She has to sneak around the house briefly before she figures out that it’s in the desk by the door. If you can grab it first, then congratulations. On midnight tomorrow you can turn it back in with Armand to gain access to the first fence and join the Thieves Guild.
If you didn't get Amantius Allectus' Diary, then just read more on the next page for you new options.