How did Dumbledore know where to find Slughorn in Half Blood Prince?

Question

Slughorn was on the run from the Death Eaters, changing locations weekly and staying in Muggle homes to evade discovery. How then did Dumbledore know where to find him when he took Harry to recruit Slughorn back to Hogwarts as a professor?

Answer 1

Although there is not a full canon answer, it is important that Slughorn had only just arrived in the house, and that he is a little forgetful. Dumbledore points out to Slughorn that he had not created the Dark Mark, which alerted him to the fact that the house had not really been visited by Death Eaters. Slughorn responds that he knew he'd forgotten something, but wouldn't have had the time to create it before Harry and Dumbledore arrived, anyway.

Which means Slughorn only just arrived. It is not told how he arrived, but most if not all forms of wizarding transport are monitored by the Ministry. Although Slughorn would probably have tried to avoid the attention of the Ministry, he may well have slipped up. Meanwhile Dumbledore had been looking for Slughorn for a while, so he must have kept close tabs on anything betraying his presence.

Precisely how Dumbledore found Slughorn isn't specified, so as far as canon is concerned, we'll simply have to assume that it's because Dumbledore is a genius - which is more than we can say about the Death Eaters.

This last part is speculation, but it's reasonable to assume that Dumbledore has contacts in the Muggle world, who can alert him to things like burglared houses. Whereas the Death Eaters would try to either avoid Muggles or torture information out of them, Dumbledore probably has some friends who come to him, out of their own volition.

Answer 2

Knowing that Slughorn had many “contacts” in the wizarding world. He probably knew who some of them were and was able to convince one of them to tell if they knew where he was hiding. Slughorn, in his old age, likely inadvertently, let slip a clue as to where he was or where he was going to be next… come to think of it he must’ve let very specific information slip on where he would be, and when because Dumbledore wrote to Harry about a week beforehand that they would be going to attempt to make contact with Slughorn.