A square sheet of paper lies in the xy-plane with corners: (...

Prompt

A square sheet of paper lies in the xy-plane with corners: (0,0), (12,0), (12,12), (0,12). Perform the following ideal folds. The paper has zero thickness, folds are perfect, and coordinates are exact. Fold 1: Fold the right half of the sheet onto the left half along the vertical line x = 6. Fold 2: Fold the top half of the current packet down onto the bottom half along the horizontal line y = 6. The packet is now a 6 by 6 square occupying [0,6] × [0,6]. Fold 3: Fold the current 6 by 6 packet along the diagonal line from (0,6) to (6,0). Fold the upper-right triangular half, where x + y > 6, onto the lower-left triangular half, where x + y < 6. The final packet is therefore the triangular region x + y ≤ 6 inside the 6 by 6 square. Now punch a single circular hole through all layers of the final folded packet. The center of the hole is at the point (4,1), using the coordinate system of the 6 by 6 square before Fold 3. After that, unfold the paper completely. Question: List the exact coordinates of all distinct hole centers in the original 12 by 12 square.