This is part four in a six-part series. If you haven't read the others yet, you should do so before reading this.
Read part one, which covers chapters 1-5,
here.
Chapter 16
A VERY FROSTY CHRISTMAS
Ron and Harry spend Christmas at the Burrow with the Weasley family and Lupin. Lupin reveals how he first became a werewolf and who is responsible.
Percy stops in for a short visit with the new M.O.M. who tries to coerce Harry into giving an appearance that he is supporting the Ministry.
Chapter 17
A SLUGGISH MEMORY
The Ministry is offering Apparition lessons to sixth years, and many students, including the trio, sign up.
Harry has his third meeting with Dumbledore, where they discuss Tom Riddle's first year at Hogwarts and use the pensieve to view the memory of a certain Gaunt relative. Through this, Harry discovers how Voldemort killed his muggle father.
The next memory they view with the pensieve belongs to Slughorn, who was Tom Riddle's potions professor. Dumbledore tells Harry that this memory is the most important, but that it has been altered. Dumbledore says Harry must persuade Slughorn to reveal Slughorns true response to Riddle asking him about 'Horcruxes'.
Chapter 18
BIRTHDAY SURPRISES
Harry is one of Slughorn's favorite students, thanks in part to the Half-Blood Prince's footnotes in Harry's potions book. Following the Prince's scribbled instructions has made Harry the best in his class.
In spite of Harry's multiple attempts, Slughorn sticks to the altered version of his memory regarding Horcruxes, which are of very dark magic. So dark, that none of Hogwarts' library books even describe what they are.
Harry starts using the Marauders Map to keep an eye on Draco Malfoy, who keeps disappearing off the map, much to Harry's dismay.
Chapter 19
ELF TAILS
Hagrid overhears Dumbledore arguing with Snape about something, and asking Snape to make investigations in his house. Harry wonders if this means that Dumbledore doesn't really trust Snape as much as he pretends to, and figures Dumbledore wanted Malfoy watched closer.
Becoming ever more suspicious of Malfoy, Harry assigns a couple of house-elves to follow Draco around and see what he's up to, givinig Harry regular reports.
Chapter 20
LORD VOLDEMORT'S REQUEST
Harry meets Dumbledore for their fourth traning session, and must confess that he hasn't yet managed to get the unaltered memory about the Horcruxes from Slughorn. Dumbledore is disappointed, and says that this will then be their last meeting until Harry accomplishes that task.
Using the pensieve, Dumbledore shows Harry two memories. The first one reveals Voldemort murdering once more - this time to colect some precious heirlooms. The second, one of Dumbledore's own memories regarding Voldemort. It is 10 years after Tom Riddle graduates from Hogwarts, and he returns to Dumbledore, who is now Headmaster, to ask for employment.
Continue to part five (chapters 21-25) 
HBP Chapter-By-Chapter: 16-20 (mild spoilers) 
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