Question

A rectangular wooden plank has another smaller rectangular hole cut into it as shown in the figure given below. What is the minimum number of cutting planes that can divide the wooden plank into four parts of equal volume?

Answer 1

Correct Answer: 2

To solve this problem, we need to determine the minimum number of cutting planes required to divide a rectangular wooden plank with a smaller rectangular hole into four parts of equal volume. Consider the following steps: 

1. Understanding the Structure: We have a wooden plank with a hole, and our objective is to create four parts of equal volume. Each part must be a solid piece.
2. Visualize Cutting Planes: Use three-dimensional reasoning to visualize potential cutting planes. Cutting planes can be oriented along the length, width, or height of the plank.
3. Determine Symmetrical Cuts: Analyze how symmetrical cuts can be used. Start with the hole-free parts of the plank to ensure symmetry and equality in volume.
4. Plan the Cuts: Consider that we can make two cuts parallel to the plank's plane without reducing symmetry:
   • First Plane: Divide the plank lengthwise into two equal sections.
   • Second Plane: Divide these sections widthwise into equal parts to form four parts.

Conclusion: Only two cutting planes are necessary to achieve four parts of equal volume. Both cuts leverage the symmetry of the hole in the wooden plank.

Verification: The calculated solution is verified against the expected range (2,2), confirming that the solution is efficient and satisfies the conditions without excess cuts.