4. A square piece of paper is cut into two parts along a straight line that does not pass through any vertex; take one of the parts and cut it into two parts along a straight line that does not pass through any vertex; then take one of the resulting three parts and cut it into two parts along a straight line that does not pass through any vertex... and so on. In the end, 34 hexagons and some other polygonal pieces are obtained. The minimum number of cuts required is ( ).
(A) 2004
(B) 2005
(C) 2006
(D) 2007
4. B.

According to the problem, when cutting along a line that does not pass through the vertices with the front knife, each cut increases the sum of the interior angles of each part by \(360^{\circ}\). Thus, after \(k\) cuts, \(k+1\) polygons can be obtained, and the sum of the interior angles of these polygons is \((k+1) \times 360^{\circ}\).

Since among these \(k+1\) polygons, there are 34 hexacontakaihexagons (62-sided polygons), their sum of interior angles is
\[
34 \times (62-2) \times 180^{\circ} = 34 \times 60 \times 180^{\circ}.
\]

The remaining polygons are \((k+1)-34 = k-33\) in number, and the sum of their interior angles is no less than \((k-33) \times 180^{\circ}\). Therefore,
\[
(k+1) \times 360^{\circ} \geqslant 34 \times 60 \times 180^{\circ} + (k-33) \times 180^{\circ}.
\]

Solving this, we get \(k \geqslant 2005\).
When cutting 2005 times in the following manner, the required result can be achieved: First, cut 1 triangle from a square, resulting in 1 quadrilateral and 1 pentagon; then cut 1 triangle from the pentagon, resulting in 2 triangles and 1 hexagon... After 58 cuts, 58 triangles and 1 hexacontakaihexagon (62-sided polygon) are obtained. Next, take out 33 triangles, and cut each one, resulting in 33 triangles and 33 quadrilaterals. For these 33 quadrilaterals, use the same cutting method as for the square, and cut each one 58 times, resulting in 33 hexacontakaihexagons (62-sided polygons) and \(33 \times 58\) triangles. Thus, a total of
\[
58 + 33 + 33 \times 58 = 2005 \text{ (cuts)}.
\]