AMC 8 Daily Practice Round 9

Complete problem set with solutions and individual problem pages

Problem 18 Hard

A total of 25 unit cubes, each with an edge length of 1, are stacked together to form a solid shape. What is the minimum possible surface area of the resulting geometric figure?

  • A.

    50

  • B.

    54

  • C.

    56

  • D.

    70

  • E.

    75

Answer:B

The surface area is minimized when the unit cubes overlap as much as possible.

Consider a cube composed of 27 unit cubes (3 \times 3 \times 3), which has the smallest possible surface area when fully assembled.

To reduce the total count to 25 unit cubes, we need to remove 2 cubes in a way that does not increase the surface area. This can be achieved by removing:

1. Two cubes from two different corners, or

2. Two adjacent cubes from the same corner.

In both cases, the surface area remains unchanged at 54.

Thus, the minimum possible surface area of the resulting shape is 54.

The answer is \text{B}.

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