Unfold through overlapping layers

Begin with an 8 × 8 square. Fold its lower-left half across the corner-to-corner diagonal onto the upper-right half. The two triangular halves now overlap. Each interior position in that triangle has two paper layers.

Next fold the paper to the left across the vertical middle line. Some moving paper lands on the existing triangle; some lands where there was no paper. The strongly shaded upper region has four layers, while the rest has two. The dot marks the punch, dashed lines mark creases, and the grid helps locate positions.

A punch cuts only through the layers directly beneath it. To unfold, undo the last fold first, moving the holes with their own pieces of paper. Do not reflect every hole and keep both copies unless paper existed on both sides at that position.

Punch at row 6, column 2 of the original grid, counting from the top-left. This point lies below the four-layer overlap, so the punch goes through only two layers.

Undo the vertical fold first. Both punched layers move right to row 6, column 7, giving one visible hole position in the still-folded triangle. Do not leave a hole at row 6, column 2: before the vertical fold, that position was outside the triangle and held no paper.

Now open the diagonal fold. Row 6, column 7 reflects to row 7, column 6. There are exactly two holes in the open square. This diagonal exchanges row and column. Check by folding both positions forward again: they meet at the original punch.

Use the same diagonal fold followed by the same leftward vertical fold. This time punch at row 2, column 3. Only the punch position changes.

Check whether the moving layers landed on paper already present at this point. Undo the vertical fold, then the diagonal. Count distinct holes on the fully opened sheet, not the number of dots visible while folded.