Question

The front view of a solid object is shown in the image. If the views from all six sides are the same, how many surfaces does it have?

Hint:

In spatial reasoning problems where an object has identical views from all sides, break it down into a central body and identical radiating arms. Calculate the surfaces for one arm and multiply by the number of arms (usually 6 for cubic symmetry).

Answer 1

Correct Answer: 72

Step 1: Identify the repeating layer types.
Each of the 6 arms is built the same way: an inner cube next to the centre, a middle cube next to that, and a pyramid tip, so every arm has 3 layers of the same kind.

Step 2: Count exposed faces per layer type, across all arms at once.
Each inner cube shows 4 side faces (its other two faces are glued to the centre and the middle cube), so all 6 inner cubes together show \(6 \times 4 = 24\) faces.
Each middle cube also shows only its 4 side faces, giving another \(6 \times 4 = 24\) faces.
Each pyramid shows its 4 triangular faces (its square base is glued to the middle cube), giving \(6 \times 4 = 24\) faces.

Step 3: Add the layer totals.
\[ 24 + 24 + 24 = 72 \]

Step 4: Final Answer.
The object has \(\boxed{72}\) surfaces.

Answer 2

A quicker way to reach the same total is to notice that each of the 6 arms is just a straight stack of shapes glued end to end, and a stack like that has a simple rule for how many faces stay visible.

Whenever two solids are glued face to face inside a stack, that shared face disappears from both solids, so only the "side" faces of each cube in the stack, and the slanted faces of a pyramid capping the stack, remain visible.
For a cube inside a stack (not at either end), only its 4 side faces are exposed, since its top and bottom faces are glued to the pieces before and after it in the stack.
Each arm here is a stack of 2 cubes followed by a pyramid cap. Both cubes sit mid-stack (each has one face glued to a neighbouring piece on either side), so each contributes 4 visible side faces: \[ 2 \times 4 = 8 \text{ faces from the two cubes} \]
The pyramid caps the stack, so only its 1 glued base disappears and its 4 triangular faces stay visible, adding 4 more faces. \[ 8 + 4 = 12 \text{ faces per arm} \]
Since the object has 6 identical arms attached to the hidden central cube (whose own faces are all covered), the total number of visible faces is \[ 6 \times 12 = 72 \]

So the correct answer is 72.