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?
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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).
Step 1: Set up the rule.
Every piece starts with all its own faces exposed. Gluing it to a neighbour hides one face on it and one face on that neighbour. So for any piece, exposed faces equal its own face count minus the number of neighbours it is glued to.
Step 2: Apply the rule to the central cube.
The central cube has 6 faces and is glued to all 6 inner cubes, one per arm, so its neighbour count is 6.
\[ 6 - 6 = 0 \]
The central cube contributes 0 exposed faces, since it is completely covered.
Step 3: Apply the rule inside one arm.
The inner cube, 6 faces, is glued to 2 neighbours, the central cube and the middle cube: \( 6 - 2 = 4 \) exposed faces.
The middle cube, 6 faces, is glued to 2 neighbours, the inner cube and the pyramid: \( 6 - 2 = 4 \) exposed faces.
The pyramid has 5 faces (1 square base plus 4 triangles) and is glued to 1 neighbour at its base: \( 5 - 1 = 4 \) exposed faces.
So one arm gives \( 4 + 4 + 4 = 12 \) exposed faces.
Step 4: Scale up to all six arms.
\[ \text{Total} = 0 + 6 \times 12 = 72 \]
\[ \boxed{72} \]
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Approach Solution -2
A third way to reach the same count is to sort every exposed face by its shape, square or triangular, instead of working piece by piece or joint by joint.
Squares come only from the side faces of the inner and middle cubes, since the central cube is fully hidden and the pyramid's base is glued down. Each inner cube shows 4 of its 6 faces (the two faces along the arm's direction are glued), so 6 inner cubes give 6 x 4 = 24 square faces. Each middle cube also shows 4 faces for the same reason, giving another 6 x 4 = 24 square faces. That is 48 square faces in total.
Triangles come only from the pyramids. Each pyramid shows 4 triangular side faces, since its square base is glued to the middle cube, so 6 pyramids give 6 x 4 = 24 triangular faces.
Adding the two groups, 48 square faces plus 24 triangular faces gives 72 faces in all.
Sorting the faces by shape gives the same total as before, so the correct answer is 72.
