A cube is created by stacking 27 smaller cubes as shown in the figure. A plane, going through vertices A, B and C, cuts the cube as shown in the figure. How many smaller cubes will get cut?
To solve the problem of determining how many smaller cubes are cut by the plane, we need to consider the arrangement and properties of the cube. The large cube is composed of 27 smaller cubes, structured as a 3x3x3 grid.
Given that the plane passes through three vertices A, B, and C, and divides the cube, our task is to identify how many of these smaller cubes are intersected by the plane.
This plane effectively slices through the cube diagonally from corner to corner. Since the cube consists of 3 layers of 3x3 smaller cubes, we'll analyze each layer separately:
In the bottom layer, the plane cuts through a diagonal row of cubes.
This pattern repeats in the middle and top layers.
The plane follows a diagonal from vertex to vertex, cutting through one cube per row per layer. Considering all axes are equally bisected due to the symmetrical nature of the cube, the plane cuts one smaller cube from each row it intersects.
With each layer contributing one diagonal line of cubes, and each line containing three cubes (one from each row in the grid per layer), this plane cuts through 3 cubes per layer.
Thus, across all three layers of the cube, the plane cuts:
3 (cubes per layer) × 3 (layers) = 9 cubes
This ensures all cubes intersected by the plane are accounted for.
The computed value of 9 delicate cubes is indeed within the provided range of 9,9, corroborating our analysis.
