Comment by cubefox

Gemini did not one-shot these answers; it did its thinking elsewhere (probably not released by Google) and then it consolidated it down into what you see in the PDF. From the article:

> We achieved this year’s result using an advanced version of Gemini Deep Think – an enhanced reasoning mode for complex problems that incorporates some of our latest research techniques, including parallel thinking. This setup enables the model to simultaneously explore and combine multiple possible solutions before giving a final answer, rather than pursuing a single, linear chain of thought.

I don't see any parallel thinking, e.g., so that was probably elided in the final results.

It's not it being case by case that's my issue. I used do olympiads and e.g. for the k>=3 case I wouldn't write much more than:

"Since there are 3k - 3 points on the perimeter of the triangle to be covered, and any sunny line can pass through at most two of them, it follows that 3k − 3 ≤ 2k, i.e. k ≤ 3."

Gemini writes:

Let Tk be the convex hull of Pk. Tk is the triangle with vertices V1 = (1, 1), V2 = (1, k), V3 = (k, 1). The edges of Tk lie on the lines x = 1 (V), y = 1 (H), and x + y = k + 1 (D). These lines are shady.

Let Bk be the set of points in Pk lying on the boundary of Tk. Each edge contains k points. Since the vertices are distinct (as k ≥ 2), the total number of points on the boundary is |Bk| = 3k − 3.

Suppose Pk is covered by k sunny lines Lk. These lines must cover Bk. Let L ∈ Lk. Since L is sunny, it does not coincide with the lines containing the edges of Tk. A line that does not contain an edge of a convex polygon intersects the boundary of the polygon at most at two points. Thus, |L ∩ Bk| ≤ 2. The total coverage of Bk by Lk is at most 2k. We must have |Bk| ≤ 2k. 3k − 3 ≤ 2k, which implies k ≤ 3.