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Zorluk: Çok zorCubes and Dice

A large solid cube is constructed using 125125 smaller, identical cubic blocks. Before the large cube is painted, all 88 corner blocks are completely removed from the structure. The resulting modified structure is then completely dipped in a vat of red paint. After the paint dries, the structure is disassembled back into the remaining 117117 individual smaller blocks. How many of these 117117 smaller blocks have exactly two of their faces painted red?

  1. 12Cevap
  2. B
    24
  3. C
    36
  4. D
    60

Cevap

12
In a standard 5×5×55 \times 5 \times 5 cube, the blocks with exactly two painted faces are located along the edges (excluding the corners). There are 5−2=35 - 2 = 3 such blocks per edge. However, when the 88 corner blocks are removed, the structural integrity changes. The edge blocks directly adjacent to the removed corners get an additional face exposed (the inner face that was previously touching the corner). This increases their painted faces from 22 to 33. Only the single block in the exact middle of each of the 1212 edges remains unaffected by the corner removal, keeping exactly 22 exposed faces. Thus, there are 12 edges×1 block/edge=1212 \text{ edges} \times 1 \text{ block/edge} = 12 such blocks.

Adım Adım Çözüm

1
Analyze the original edge composition of the 5x5x5 cube.
The cube has 12 edges. Each edge consists of 5 blocks: 2 corner blocks and 3 middle edge blocks.
To establish the baseline before the corners are removed.
2
Determine the spatial effect of removing the 8 corner blocks.
Removing a corner creates a 'dent'. The 3 edge blocks that were directly touching this corner now have an interior face exposed to the outside.
Paint will now reach surfaces that were previously hidden inside the structure.
3
Calculate the newly painted faces for the affected edge blocks.
The 2 edge blocks at the ends of each 3-block edge segment originally had 2 exposed faces. They now have 1 additional exposed face, bringing their total to 3 painted faces.
To eliminate these blocks from the target count of blocks with exactly 2 painted faces.
4
Identify the blocks that maintain exactly two exposed faces.
Only the single block in the exact center of each 5-block edge does not touch a corner. Its exposed faces remain unchanged at 2.
These are the only blocks in the entire modified structure that fit the criteria.
5
Calculate the final total.
12 edges multiplied by 1 block per edge equals 12 blocks.
To reach the final quantitative answer.

Anahtar Kavram

Spatial reasoning and dynamic visualization of modified 3D geometric structures.
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Cubes and Dice

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