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Zorluk: Çok zorCubes and Dice

A solid wooden block has dimensions 12 cm×15 cm×18 cm12 \text{ cm} \times 15 \text{ cm} \times 18 \text{ cm}. Three of its mutually adjacent faces (which meet at a single corner) are painted Red, and the remaining three faces are painted Blue. The block is then cut into smaller, identical cubes of size 1 cm×1 cm×1 cm1 \text{ cm} \times 1 \text{ cm} \times 1 \text{ cm}. How many of these smaller cubes have AT LEAST one Red face AND at least one Blue face?

Cevap: 84

Cevap

84
The unit cubes with both colors lie exactly on the 6 boundary edges separating the Red and Blue faces. By summing the lengths of these 6 edges (12+12+15+15+18+18=9012+12+15+15+18+18 = 90) and subtracting the 6 overlapping corner cubes to correct for double-counting, we arrive at exactly 84 cubes.

Adım Adım Çözüm

1
Identify the configuration of the painted faces on the block.
The 3 Red faces share one corner, and the 3 Blue faces share the diagonally opposite corner.
To accurately locate where the Red and Blue paints meet on the surface.
2
Determine which unit cubes satisfy the condition of having both Red and Blue faces.
Only the cubes lying exactly on the boundary edges between a Red face and a Blue face will have both colors.
Cubes fully within a Red face have no Blue paint, internal cubes have no paint, and purely corner cubes not on the boundary have only one color.
3
Count the number of boundary edges.
There are exactly 6 boundary edges forming a continuous zig-zag hexagonal ring around the block.
Each of the 3 Red faces shares an edge with exactly 2 Blue faces.
4
Determine the lengths of these 6 boundary edges.
The ring wraps around all three dimensions symmetrically, covering two edges of 12 cm, two of 15 cm, and two of 18 cm.
A cuboid has 4 edges of each dimension; the boundary ring traverses exactly half of them.
5
Calculate the total number of unit cubes on these edges using the inclusion-exclusion principle.
2×(12+15+18)−6=90−6=842 \times (12 + 15 + 18) - 6 = 90 - 6 = 84 cubes.
Summing the edge lengths gives 90 cubes, but the 6 corners connecting these edges in the ring are counted twice, so we must subtract 6 to prevent double-counting.

Anahtar Kavram

Spatial reasoning and painted cube boundary edge calculation.
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Cubes and Dice

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