‘You have to actually think’: How to solve the hardest HSC maths problem

Inside the small club of students across NSW who take on the hardest level of Higher School Certificate mathematics, certain papers are notorious for their difficulty.

There was the final locus problem in 2023. The 2001 paper’s last question is still discussed on obscure parts of the internet.

When North Sydney Girls High School student Angela Lin, 18, woke up on her birthday on Monday, she was feeling comfortable about the exam she would sit alongside 3769 NSW students.

But then she received word that the mathematics advanced paper that morning was extremely difficult, and the stress over extension 2 began. But once the exam started, that anxiety subsided.

“I saw a lot of diagrams, and I was like, ‘OK … I’m probably going to have to slow down and understand those first’,” she said.

The final and most difficult question, 16C, was on vectors, with a diagram of a sphere similar to the 2025 exam.

In mathematics, a vector is a quantity which has length or size, coupled with direction.

It had its challenges, but it was not a repeat of the difficulty faced by students in 2023.

“I thought it was a pretty good paper. Especially the last question,” she said.

The problem was split into three parts. The first section asked students to prove that the vector had a length of 1. The second asked them to find intersections between two circles in three-dimensional space, by solving two vector equations simultaneously.

The final stage was more difficult, requiring students to make connections between the information provided in the question and vector projections, to derive the vector equation of a circle in 3D space passing through two given points.

For Annie Chen, 18, the lack of any odd shapes or eccentric graphs in the paper was welcomed, but the last question threw up a few curveballs.

“There were hiccups with trying to relate again to the last part, to the previous parts of the question. I think that requires a little bit of thinking,” she said.

The backdrop to Monday’s HSC exam is a mathematics world is in turmoil.