"He's a pureblood!" "His surname is Williams!" "He has 4 magical grandparents! It doesn't matter that all 4 of them are mudbloods, he's still a pureblood by definition!"
Richard Williams. Born in Manchester, sorted into Ravenclaw. A perfectly ordinary student. The only way he could be more unassuming is if he were sorted in Hufflepuff.
However, there was something odd about Richard. Both of his parents were squibs, all 4 of his grandparents were wizards, and all 8 of his great-grandparents were muggles.
In a less bigoted world, his family tree would be nothing more than a quirky statistical anomaly.
In this world however, his family history was forced out of him after a few upper-year slytherins tried to bully him.
Initially, they didn't give it much thought. A little mudblood trying to come up with a reason to be left alone.
The infighting would come later.
"He should count, no?"
"Of course he shouldn't count! You're not a pureblood if you know how closely related to a muggle you are!"
"By that criteria, we'd have to exclude 9 out of 10 people in Slytherin! Weren't the Malfoys muggle nobility?"
It didn't happen overnight, but the rift of the pureblood faction began that day. Not by moral improvement of its members, but by truly thinking about the definition of an inherently unscientific idea.