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Difficulty: HardCubes and Dice

A large solid cube of dimensions 5×5×55 \times 5 \times 5 is painted using three colors: Red, Blue, and Green. The painting is applied such that the Top and Bottom faces are Red, the Front and Right faces are Blue, and the Back and Left faces are Green. The large cube is then cut into 125 smaller, identical cubes of dimensions 1×1×11 \times 1 \times 1. Match the categories of the smaller cubes described in List I with their exact counts in List II.

  • Cubes with exactly three faces painted, displaying three different colors4
  • Cubes with exactly two faces painted, both displaying the same color6
  • Cubes with exactly one face painted Blue18
  • Cubes with no faces painted at all27

Answer

The correct matches are: Cubes with three different colors to 4; Cubes with two faces of the same color to 6; Cubes with one face Blue to 18; Cubes with no painted faces to 27.
The exact structural breakdown of the 5x5x5 cube under this specific adjacent-face coloring scheme yields exactly 4 three-color corners, 6 two-color edges (from 2 identical color intersections), 18 one-color blue face centers, and a 27-cube unpainted core.

Step-by-Step Solution

1
Analyze the corner cubes (exactly 3 faces painted) to match the first category.
There are 8 corners. Top-Front-Left is Red-Blue-Green (3 colors). Top-Front-Right is Red-Blue-Blue (2 colors). Analyzing all 8 corners yields exactly 4 corners that touch three different colors.
Because Front/Right are both Blue and Back/Left are both Green, corners sharing these adjacent identical faces will not have three different colors.
2
Analyze the edge cubes (exactly 2 faces painted) to match the second category.
There are 12 edges, each containing (5-2) = 3 cubes with exactly two painted faces. The condition requires both faces to be the same color. This only happens on the Front-Right edge (Blue-Blue) and Back-Left edge (Green-Green). Thus, 2 edges × 3 cubes = 6 cubes.
Edges represent the intersection of two faces. Only adjacent faces painted with the same color produce these specific cubes.
3
Analyze the face center cubes (exactly 1 face painted) to match the third category.
Each of the 6 faces has a central grid of (5-2)² = 9 cubes that have exactly one face painted. Since exactly 2 faces are painted Blue (Front and Right), there are 2 × 9 = 18 such cubes.
This singles out the purely Blue surface area minus edges and corners.
4
Calculate the completely unpainted inner core cubes to match the fourth category.
The unpainted inner volume is a smaller cube of dimensions (5-2) × (5-2) × (5-2) = 3³ = 27 cubes.
Removing the outer layer (1 cube thick) from all 6 sides leaves the unpainted core.

Key Concept

Advanced visualization of 3D spatial properties, edge intersections, and non-standard coloring rules on a subdivided cube.
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Topics

Cubes and Dice

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