DAT Quiz: Analyze 3d Cubes
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Analyze 3d CubesQuestion 1 of 20

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In Figure 1 as shown, how many cubes have exactly five of their exposed sides painted?

Question graphic
1 cube
2 cubes
3 cubes
4 cubes
5 cubes
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DAT Quiz: Analyze 3d Cubes

Practice Analyze 3d Cubes in DAT with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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Question 1

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In Figure 1 as shown, how many cubes have exactly five of their exposed sides painted?

  1. 1 cube (correct answer)
  2. 2 cubes
  3. 3 cubes
  4. 4 cubes
  5. 5 cubes
Explanation: Figure 1 Explanation: Spatial Logic and Painted Sides To determine the number of painted sides, each cube must be analyzed by its position within the structure and the number of faces blocked by the ground (bottom) or neighboring cubes (top, front, back, left, or right) Level 1 (Base Layer: 8 Cubes)T he base layer is a 3×33 \times 3 grid with the front-left corner omitted. 1 Side Painted (3 Cubes):The Center Cube: Blocked by the ground and all four horizontal neighbors. Its top is the only exposed face.The Back-Middle and Middle-Left Cubes: These are blocked by the ground, three horizontal neighbors, and the Level 2 cubes resting directly on top of them. Only one outer vertical face is exposed for each. 2 Sides Painted (1 Cube):The Back-Left Corner: Blocked by the ground, the Level 2 corner cube above it, and two horizontal neighbors. Only the back and left vertical faces are exposed. 3 Sides Painted (4 Cubes):The Outer Corners and Edges: The remaining base cubes have their tops exposed along with two outer vertical faces. These include the back-right corner, the front-right corner, and the two edges bordering the "void" where the front-left corner was removed. Level 2 (Second Layer: 3 Cubes) These three cubes form an "L" shape directly above the back-left corner of the base. 2 Sides Painted (1 Cube):The Corner Cube: Blocked on the bottom (by Level 1), the top (by Level 3), and two horizontal sides (by its Level 2 "wings"). Only the back and left vertical faces are exposed. 4 Sides Painted (2 Cubes):The "Wings": These cubes are blocked only on the bottom and the one side connected to the corner. This leaves the top and three vertical faces exposed. Level 3 (Third Layer: 1 Cube) 4 Sides Painted (1 Cube):This single cube is blocked on the bottom (by Level 2) and the top (by Level 4). All four vertical faces are exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube is blocked only on its bottom side. The top and all four vertical faces are exposed.

Question 2

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In FIgure 1 as shown, how many cubes have exactly four of their exposed sides painted?

  1. 1 cube
  2. 2 cubes
  3. 3 cubes (correct answer)
  4. 4 cubes
  5. 5 cubes
Explanation: Figure 1 Explanation: Spatial Logic and Painted Sides To determine the number of painted sides, each cube must be analyzed by its position within the structure and the number of faces blocked by the ground (bottom) or neighboring cubes (top, front, back, left, or right) Level 1 (Base Layer: 8 Cubes)T he base layer is a 3×33 \times 3 grid with the front-left corner omitted. 1 Side Painted (3 Cubes):The Center Cube: Blocked by the ground and all four horizontal neighbors. Its top is the only exposed face.The Back-Middle and Middle-Left Cubes: These are blocked by the ground, three horizontal neighbors, and the Level 2 cubes resting directly on top of them. Only one outer vertical face is exposed for each. 2 Sides Painted (1 Cube):The Back-Left Corner: Blocked by the ground, the Level 2 corner cube above it, and two horizontal neighbors. Only the back and left vertical faces are exposed. 3 Sides Painted (4 Cubes):The Outer Corners and Edges: The remaining base cubes have their tops exposed along with two outer vertical faces. These include the back-right corner, the front-right corner, and the two edges bordering the "void" where the front-left corner was removed. Level 2 (Second Layer: 3 Cubes) These three cubes form an "L" shape directly above the back-left corner of the base. 2 Sides Painted (1 Cube):The Corner Cube: Blocked on the bottom (by Level 1), the top (by Level 3), and two horizontal sides (by its Level 2 "wings"). Only the back and left vertical faces are exposed. 4 Sides Painted (2 Cubes):The "Wings": These cubes are blocked only on the bottom and the one side connected to the corner. This leaves the top and three vertical faces exposed. Level 3 (Third Layer: 1 Cube) 4 Sides Painted (1 Cube):This single cube is blocked on the bottom (by Level 2) and the top (by Level 4). All four vertical faces are exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube is blocked only on its bottom side. The top and all four vertical faces are exposed.

Question 3

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In Figure 1 as shown, how many cubes have exactly two of their exposed sides painted?

  1. 1 cube
  2. 2 cubes (correct answer)
  3. 3 cubes
  4. 4 cubes
  5. 5 cubes
Explanation: Figure 1 Explanation: Spatial Logic and Painted Sides To determine the number of painted sides, each cube must be analyzed by its position within the structure and the number of faces blocked by the ground (bottom) or neighboring cubes (top, front, back, left, or right) Level 1 (Base Layer: 8 Cubes)T he base layer is a 3×33 \times 3 grid with the front-left corner omitted. 1 Side Painted (3 Cubes):The Center Cube: Blocked by the ground and all four horizontal neighbors. Its top is the only exposed face.The Back-Middle and Middle-Left Cubes: These are blocked by the ground, three horizontal neighbors, and the Level 2 cubes resting directly on top of them. Only one outer vertical face is exposed for each. 2 Sides Painted (1 Cube):The Back-Left Corner: Blocked by the ground, the Level 2 corner cube above it, and two horizontal neighbors. Only the back and left vertical faces are exposed. 3 Sides Painted (4 Cubes):The Outer Corners and Edges: The remaining base cubes have their tops exposed along with two outer vertical faces. These include the back-right corner, the front-right corner, and the two edges bordering the "void" where the front-left corner was removed. Level 2 (Second Layer: 3 Cubes) These three cubes form an "L" shape directly above the back-left corner of the base. 2 Sides Painted (1 Cube):The Corner Cube: Blocked on the bottom (by Level 1), the top (by Level 3), and two horizontal sides (by its Level 2 "wings"). Only the back and left vertical faces are exposed. 4 Sides Painted (2 Cubes):The "Wings": These cubes are blocked only on the bottom and the one side connected to the corner. This leaves the top and three vertical faces exposed. Level 3 (Third Layer: 1 Cube) 4 Sides Painted (1 Cube):This single cube is blocked on the bottom (by Level 2) and the top (by Level 4). All four vertical faces are exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube is blocked only on its bottom side. The top and all four vertical faces are exposed.

Question 4

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted

The figure shown is comprised of 20 cubes: the 13 explicitly pictured and the cubes necessary to support them. Of those 20 cubes, how many have exactly five sides painted?

  1. No cubes
  2. 1 cube (correct answer)
  3. 2 cubes
  4. 3 cubes
  5. 4 cubes
Explanation: This structure contains exactly 20 identical cubes arranged in a tiered V-shape with a four-story central tower. The analysis determines the number of painted sides for each cube by calculating its exposed surfaces, excluding the bottom surface resting on the ground. 0 Sides Painted (3 Cubes):These are the three "landlocked" support cubes at the base of the inner V. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and neighbors on all four lateral sides. 1 Side Painted (0 Cubes):In this specific structure, no cube has exactly one face exposed. Every cube either has zero exposure (if landlocked) or at least two faces peering out. 2 Sides Painted (8 Cubes)The Four-Story Tower Base: The bottom two cubes of the central tower are blocked on the top/bottom and two side neighbors. This leaves exactly two vertical faces exposed per cube .Middle Arm Supports (L1): The two cubes at the base of the Level 2 "middle arms" are blocked by ground, the cube above, and two lateral neighbors.Middle Arm Visible Blocks (L2): These two visible blocks are blocked on the bottom and two sides. With the top exposed, they have two faces painted (Top and Front).End Arm Supports (L1): The two cubes hidden beneath the Level 2 wingtips are blocked on the top, bottom, and two sides. This leaves the front and back faces exposed. 3 Sides Painted (5 Cubes)Bottom Row End Pieces: The four single-story cubes at the very ends of the base V have their top and two outer vertical faces exposed .Second Row Vertex: The cube at the vertex of the Level 2 V is blocked on the bottom and two lateral sides. This leaves the top, front, and back faces exposed. 4 Sides Painted (3 Cubes)Level 2 Wingtips: The two cubes at the far ends of the Level 2 V are blocked only on the bottom and one lateral side. This leaves the top and three vertical faces exposed.Third Level Tower Block: This cube is blocked on the top and bottom by other tower segments. All four of its vertical sides are exposed . 5 Sides Painted (1 Cube)The Topmost Cube: The cube at the very peak (Level 4) is blocked only on its bottom surface where it sits on Level 3. All five other faces are painted .

Question 5

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted

The figure shown is comprised of 20 cubes: the 13 explicitly pictured and the cubes necessary to support them. Of those 20 cubes, how many have exactly two sides painted?

  1. 4 cubes
  2. 6 cubes
  3. 7 cubes
  4. 8 cubes (correct answer)
  5. 9 cubes
Explanation: This structure contains exactly 20 identical cubes arranged in a tiered V-shape with a four-story central tower. The analysis determines the number of painted sides for each cube by calculating its exposed surfaces, excluding the bottom surface resting on the ground. 0 Sides Painted (3 Cubes):These are the three "landlocked" support cubes at the base of the inner V. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and neighbors on all four lateral sides. 1 Side Painted (0 Cubes):In this specific structure, no cube has exactly one face exposed. Every cube either has zero exposure (if landlocked) or at least two faces peering out. 2 Sides Painted (8 Cubes)The Four-Story Tower Base: The bottom two cubes of the central tower are blocked on the top/bottom and two side neighbors. This leaves exactly two vertical faces exposed per cube .Middle Arm Supports (L1): The two cubes at the base of the Level 2 "middle arms" are blocked by ground, the cube above, and two lateral neighbors.Middle Arm Visible Blocks (L2): These two visible blocks are blocked on the bottom and two sides. With the top exposed, they have two faces painted (Top and Front).End Arm Supports (L1): The two cubes hidden beneath the Level 2 wingtips are blocked on the top, bottom, and two sides. This leaves the front and back faces exposed. 3 Sides Painted (5 Cubes)Bottom Row End Pieces: The four single-story cubes at the very ends of the base V have their top and two outer vertical faces exposed .Second Row Vertex: The cube at the vertex of the Level 2 V is blocked on the bottom and two lateral sides. This leaves the top, front, and back faces exposed. 4 Sides Painted (3 Cubes)Level 2 Wingtips: The two cubes at the far ends of the Level 2 V are blocked only on the bottom and one lateral side. This leaves the top and three vertical faces exposed.Third Level Tower Block: This cube is blocked on the top and bottom by other tower segments. All four of its vertical sides are exposed . 5 Sides Painted (1 Cube)The Topmost Cube: The cube at the very peak (Level 4) is blocked only on its bottom surface where it sits on Level 3. All five other faces are painted .

Question 6

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted

The figure shown is comprised of 20 cubes: the 14 explicitly pictured and the 6 cubes necessary to support them. Of those 20 cubes, how many have zero sides painted?

  1. No cubes
  2. 1 cube
  3. 2 cubes
  4. 3 cubes (correct answer)
  5. 4 cubes
Explanation: This structure consists of 20 total cubes (14 visible and 6 inferred as necessary supports). The side counts are determined by identifying how many faces of each cube are blocked by the ground (bottom) or neighboring cubes. Level 1 (Base Layer: 12 Cubes) 0 Sides Painted (3 Cubes):These are the three "landlocked" support bricks located directly beneath the visible Level 2 wings. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and horizontal neighbors on all four sides. 2 Sides Painted (3 Cubes): Front Middle Row: The two visible cubes in the front-middle of the base are blocked by ground, their neighbors, and the structure behind them, leaving the top and front faces exposed .Tower Base: The inferred Level 1 cube at the very base of the 4-stack tower. Due to the "jagged" formation, it has two faces exposed to the exterior. 3 Sides Painted (6 Cubes): Four Corner Pieces: These are the four outermost cubes of the base (far left/right and front corners). With nothing above them, their top and two vertical faces are exposed. Back Row Partial Supports: Two partially-shown Level 1 cubes in the back have three faces exposed. Level 2 (Second Layer: 4 Cubes) 2 Sides Painted (3 Cubes):Tower Support: One inferred Level 2 cube supporting the central tower.Wing Supports: The two visible Level 2 blocks that support the 4-stack tower. They are blocked on the top, bottom, and two sides. 3 Sides Painted (1 Cube):The single Level 2 block that has nothing resting atop it. It is blocked on the bottom and two sides, leaving the top, front, and back faces exposed. Level 3 (Third Layer: 3 Cubes) 2 Sides Painted (1 Cube):The inferred Level 3 cube that supports the peak of the 4-stack tower. 4 Sides Painted (2 Cubes):The two visible Level 3 cubes at the tops of the "steps." They are blocked only on the bottom and one lateral side by the central tower, leaving the top and three vertical faces exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube of the 4-stack tower is blocked only on its bottom surface. All five other faces (top and four sides) are exposed.

Question 7

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted

The figure shown is comprised of 20 cubes: the 13 explicitly pictured and the cubes necessary to support them. Of those 20 cubes, how many have exactly three sides painted?

  1. 3 cubes
  2. 4 cubes
  3. 5 cubes (correct answer)
  4. 6 cubes
  5. 7 cubes
Explanation: This structure contains exactly 20 identical cubes arranged in a tiered V-shape with a four-story central tower. The analysis determines the number of painted sides for each cube by calculating its exposed surfaces, excluding the bottom surface resting on the ground. 0 Sides Painted (3 Cubes):These are the three "landlocked" support cubes at the base of the inner V. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and neighbors on all four lateral sides. 1 Side Painted (0 Cubes):In this specific structure, no cube has exactly one face exposed. Every cube either has zero exposure (if landlocked) or at least two faces peering out. 2 Sides Painted (8 Cubes)The Four-Story Tower Base: The bottom two cubes of the central tower are blocked on the top/bottom and two side neighbors. This leaves exactly two vertical faces exposed per cube .Middle Arm Supports (L1): The two cubes at the base of the Level 2 "middle arms" are blocked by ground, the cube above, and two lateral neighbors.Middle Arm Visible Blocks (L2): These two visible blocks are blocked on the bottom and two sides. With the top exposed, they have two faces painted (Top and Front).End Arm Supports (L1): The two cubes hidden beneath the Level 2 wingtips are blocked on the top, bottom, and two sides. This leaves the front and back faces exposed. 3 Sides Painted (5 Cubes)Bottom Row End Pieces: The four single-story cubes at the very ends of the base V have their top and two outer vertical faces exposed .Second Row Vertex: The cube at the vertex of the Level 2 V is blocked on the bottom and two lateral sides. This leaves the top, front, and back faces exposed. 4 Sides Painted (3 Cubes)Level 2 Wingtips: The two cubes at the far ends of the Level 2 V are blocked only on the bottom and one lateral side. This leaves the top and three vertical faces exposed.Third Level Tower Block: This cube is blocked on the top and bottom by other tower segments. All four of its vertical sides are exposed . 5 Sides Painted (1 Cube)The Topmost Cube: The cube at the very peak (Level 4) is blocked only on its bottom surface where it sits on Level 3. All five other faces are painted .

Question 8

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In the figure shown, how many cubes have exactly one side painted?

  1. 3 cubes
  2. 4 cubes
  3. 5 cubes
  4. 6 cubes
  5. 7 cubes (correct answer)
Explanation: This structure consists of 20 total cubes. Following the standard DAT protocol, we assume all visible cubes are supported by columns beneath them and that every exposed surface is painted except for the bottom resting on the ground. Structural Breakdown Level 1 (Base): 15 cubes (arranged in a tiered 4×44 \times 4 pattern with the front corner omitted). Level 2: 4 cubes (forming a row that supports the higher tiers). Level 3: 1 cube (the topmost peak). Painted Sides Analysis To calculate the sides, we subtract the blocked faces (ground and neighboring cubes) from the total of 6 faces per cube. 1 Side Painted (7 Cubes):Interior L1s: The four interior base cubes have their tops blocked by Level 2 and their sides blocked by neighbors, leaving only their Top exposed (if not covered) or a single outer face. Sub-supports: The three Level 1 cubes that are directly beneath exactly one other block. 2 Sides Painted (5 Cubes): Tower Base & Mid: The Level 1 and Level 2 cubes at the back corner of the tower column.Middle Pieces: Three visible Level 1 cubes that are "middle" pieces (flanked by neighbors on both sides), leaving only the top and front faces exposed. 3 Sides Painted (5 Cubes): L1 Corners: The four cubes at the outermost corners of the base layer. With nothing above them, their top and two outer faces are exposed. Non-end L2: The one visible Level 2 piece that is not on an end (blocked on bottom and two sides). 4 Sides Painted (2 Cubes): L2 End Pieces: The two visible Level 2 cubes on the ends of the row. These are blocked only on the bottom and one lateral side. 5 Sides Painted (1 Cube): The Peak: The lone Level 3 cube at the very top. Only its bottom is blocked .

Question 9

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

The figure shown is comprised of 20 cubes: the 14 explicitly pictured and the 6 cubes necessary to support them. Of those 20 cubes, how many have exactly four sides painted?

  1. No cubes
  2. 1 cube
  3. 2 cubes (correct answer)
  4. 3 cubes
  5. 4 cubes
Explanation: This structure consists of 20 total cubes (14 visible and 6 inferred as necessary supports). The side counts are determined by identifying how many faces of each cube are blocked by the ground (bottom) or neighboring cubes. Level 1 (Base Layer: 12 Cubes) 0 Sides Painted (3 Cubes):These are the three "landlocked" support bricks located directly beneath the visible Level 2 wings. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and horizontal neighbors on all four sides. 2 Sides Painted (3 Cubes): Front Middle Row: The two visible cubes in the front-middle of the base are blocked by ground, their neighbors, and the structure behind them, leaving the top and front faces exposed .Tower Base: The inferred Level 1 cube at the very base of the 4-stack tower. Due to the "jagged" formation, it has two faces exposed to the exterior. 3 Sides Painted (6 Cubes): Four Corner Pieces: These are the four outermost cubes of the base (far left/right and front corners). With nothing above them, their top and two vertical faces are exposed. Back Row Partial Supports: Two partially-shown Level 1 cubes in the back have three faces exposed. Level 2 (Second Layer: 4 Cubes) 2 Sides Painted (3 Cubes):Tower Support: One inferred Level 2 cube supporting the central tower.Wing Supports: The two visible Level 2 blocks that support the 4-stack tower. They are blocked on the top, bottom, and two sides. 3 Sides Painted (1 Cube):The single Level 2 block that has nothing resting atop it. It is blocked on the bottom and two sides, leaving the top, front, and back faces exposed. Level 3 (Third Layer: 3 Cubes) 2 Sides Painted (1 Cube):The inferred Level 3 cube that supports the peak of the 4-stack tower. 4 Sides Painted (2 Cubes):The two visible Level 3 cubes at the tops of the "steps." They are blocked only on the bottom and one lateral side by the central tower, leaving the top and three vertical faces exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube of the 4-stack tower is blocked only on its bottom surface. All five other faces (top and four sides) are exposed.

Question 10

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

The figure shown is comprised of 20 cubes: the 14 explicitly pictured and the 6 cubes necessary to support them. Of those 20 cubes, how many have exactly three sides painted?

  1. 7 cubes (correct answer)
  2. 8 cubes
  3. 9 cubes
  4. 10 cubes
  5. 11 cubes
Explanation: This structure consists of 20 total cubes (14 visible and 6 inferred as necessary supports). The side counts are determined by identifying how many faces of each cube are blocked by the ground (bottom) or neighboring cubes. Level 1 (Base Layer: 12 Cubes) 0 Sides Painted (3 Cubes):These are the three "landlocked" support bricks located directly beneath the visible Level 2 wings. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and horizontal neighbors on all four sides. 2 Sides Painted (3 Cubes): Front Middle Row: The two visible cubes in the front-middle of the base are blocked by ground, their neighbors, and the structure behind them, leaving the top and front faces exposed .Tower Base: The inferred Level 1 cube at the very base of the 4-stack tower. Due to the "jagged" formation, it has two faces exposed to the exterior. 3 Sides Painted (6 Cubes): Four Corner Pieces: These are the four outermost cubes of the base (far left/right and front corners). With nothing above them, their top and two vertical faces are exposed. Back Row Partial Supports: Two partially-shown Level 1 cubes in the back have three faces exposed. Level 2 (Second Layer: 4 Cubes) 2 Sides Painted (3 Cubes):Tower Support: One inferred Level 2 cube supporting the central tower.Wing Supports: The two visible Level 2 blocks that support the 4-stack tower. They are blocked on the top, bottom, and two sides. 3 Sides Painted (1 Cube):The single Level 2 block that has nothing resting atop it. It is blocked on the bottom and two sides, leaving the top, front, and back faces exposed. Level 3 (Third Layer: 3 Cubes) 2 Sides Painted (1 Cube):The inferred Level 3 cube that supports the peak of the 4-stack tower. 4 Sides Painted (2 Cubes):The two visible Level 3 cubes at the tops of the "steps." They are blocked only on the bottom and one lateral side by the central tower, leaving the top and three vertical faces exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube of the 4-stack tower is blocked only on its bottom surface. All five other faces (top and four sides) are exposed.

Question 11

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

The figure shown is comprised of 20 cubes: the 14 explicitly pictured and the 6 cubes necessary to support them. Of those 20 cubes, how many have exactly two sides painted?

  1. 6 cubes
  2. 7 cubes (correct answer)
  3. 8 cubes
  4. 9 cubes
  5. 10 cubes
Explanation: This structure consists of 20 total cubes (14 visible and 6 inferred as necessary supports). The side counts are determined by identifying how many faces of each cube are blocked by the ground (bottom) or neighboring cubes. Level 1 (Base Layer: 12 Cubes) 0 Sides Painted (3 Cubes):These are the three "landlocked" support bricks located directly beneath the visible Level 2 wings. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and horizontal neighbors on all four sides. 2 Sides Painted (3 Cubes): Front Middle Row: The two visible cubes in the front-middle of the base are blocked by ground, their neighbors, and the structure behind them, leaving the top and front faces exposed .Tower Base: The inferred Level 1 cube at the very base of the 4-stack tower. Due to the "jagged" formation, it has two faces exposed to the exterior. 3 Sides Painted (6 Cubes): Four Corner Pieces: These are the four outermost cubes of the base (far left/right and front corners). With nothing above them, their top and two vertical faces are exposed. Back Row Partial Supports: Two partially-shown Level 1 cubes in the back have three faces exposed. Level 2 (Second Layer: 4 Cubes) 2 Sides Painted (3 Cubes):Tower Support: One inferred Level 2 cube supporting the central tower.Wing Supports: The two visible Level 2 blocks that support the 4-stack tower. They are blocked on the top, bottom, and two sides. 3 Sides Painted (1 Cube):The single Level 2 block that has nothing resting atop it. It is blocked on the bottom and two sides, leaving the top, front, and back faces exposed. Level 3 (Third Layer: 3 Cubes) 2 Sides Painted (1 Cube):The inferred Level 3 cube that supports the peak of the 4-stack tower. 4 Sides Painted (2 Cubes):The two visible Level 3 cubes at the tops of the "steps." They are blocked only on the bottom and one lateral side by the central tower, leaving the top and three vertical faces exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube of the 4-stack tower is blocked only on its bottom surface. All five other faces (top and four sides) are exposed.

Question 12

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In the figure shown, how many cubes have exactly three sides painted?

  1. 3 cubes
  2. 4 cubes
  3. 5 cubes (correct answer)
  4. 6 cubes
  5. 7 cubes
Explanation: This structure consists of 20 total cubes. Following the standard DAT protocol, we assume all visible cubes are supported by columns beneath them and that every exposed surface is painted except for the bottom resting on the ground. Structural Breakdown Level 1 (Base): 15 cubes (arranged in a tiered 4×44 \times 4 pattern with the front corner omitted). Level 2: 4 cubes (forming a row that supports the higher tiers). Level 3: 1 cube (the topmost peak). Painted Sides Analysis To calculate the sides, we subtract the blocked faces (ground and neighboring cubes) from the total of 6 faces per cube. 1 Side Painted (7 Cubes):Interior L1s: The four interior base cubes have their tops blocked by Level 2 and their sides blocked by neighbors, leaving only their Top exposed (if not covered) or a single outer face. Sub-supports: The three Level 1 cubes that are directly beneath exactly one other block. 2 Sides Painted (5 Cubes): Tower Base & Mid: The Level 1 and Level 2 cubes at the back corner of the tower column.Middle Pieces: Three visible Level 1 cubes that are "middle" pieces (flanked by neighbors on both sides), leaving only the top and front faces exposed. 3 Sides Painted (5 Cubes): L1 Corners: The four cubes at the outermost corners of the base layer. With nothing above them, their top and two outer faces are exposed. Non-end L2: The one visible Level 2 piece that is not on an end (blocked on bottom and two sides). 4 Sides Painted (2 Cubes): L2 End Pieces: The two visible Level 2 cubes on the ends of the row. These are blocked only on the bottom and one lateral side. 5 Sides Painted (1 Cube): The Peak: The lone Level 3 cube at the very top. Only its bottom is blocked .

Question 13

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In the figure shown, how many cubes have exactly four sides painted?

  1. 1 cube
  2. 2 cubes (correct answer)
  3. 3 cubes
  4. 4 cubes
  5. 5 cubes
Explanation: This structure consists of 20 total cubes. Following the standard DAT protocol, we assume all visible cubes are supported by columns beneath them and that every exposed surface is painted except for the bottom resting on the ground. Structural Breakdown Level 1 (Base): 15 cubes (arranged in a tiered 4×44 \times 4 pattern with the front corner omitted). Level 2: 4 cubes (forming a row that supports the higher tiers). Level 3: 1 cube (the topmost peak). Painted Sides Analysis To calculate the sides, we subtract the blocked faces (ground and neighboring cubes) from the total of 6 faces per cube. 1 Side Painted (7 Cubes):Interior L1s: The four interior base cubes have their tops blocked by Level 2 and their sides blocked by neighbors, leaving only their Top exposed (if not covered) or a single outer face. Sub-supports: The three Level 1 cubes that are directly beneath exactly one other block. 2 Sides Painted (5 Cubes): Tower Base & Mid: The Level 1 and Level 2 cubes at the back corner of the tower column.Middle Pieces: Three visible Level 1 cubes that are "middle" pieces (flanked by neighbors on both sides), leaving only the top and front faces exposed. 3 Sides Painted (5 Cubes): L1 Corners: The four cubes at the outermost corners of the base layer. With nothing above them, their top and two outer faces are exposed. Non-end L2: The one visible Level 2 piece that is not on an end (blocked on bottom and two sides). 4 Sides Painted (2 Cubes): L2 End Pieces: The two visible Level 2 cubes on the ends of the row. These are blocked only on the bottom and one lateral side. 5 Sides Painted (1 Cube): The Peak: The lone Level 3 cube at the very top. Only its bottom is blocked .

Question 14

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In the figure shown, how many cubes five exactly four sides painted?

  1. 1 cube (correct answer)
  2. 2 cubes
  3. 3 cubes
  4. 4 cubes
  5. 5 cubes
Explanation: This structure consists of 20 total cubes. Following the standard DAT protocol, we assume all visible cubes are supported by columns beneath them and that every exposed surface is painted except for the bottom resting on the ground. Structural Breakdown Level 1 (Base): 15 cubes (arranged in a tiered 4×44 \times 4 pattern with the front corner omitted). Level 2: 4 cubes (forming a row that supports the higher tiers). Level 3: 1 cube (the topmost peak). Painted Sides Analysis To calculate the sides, we subtract the blocked faces (ground and neighboring cubes) from the total of 6 faces per cube. 1 Side Painted (7 Cubes):Interior L1s: The four interior base cubes have their tops blocked by Level 2 and their sides blocked by neighbors, leaving only their Top exposed (if not covered) or a single outer face. Sub-supports: The three Level 1 cubes that are directly beneath exactly one other block. 2 Sides Painted (5 Cubes): Tower Base & Mid: The Level 1 and Level 2 cubes at the back corner of the tower column.Middle Pieces: Three visible Level 1 cubes that are "middle" pieces (flanked by neighbors on both sides), leaving only the top and front faces exposed. 3 Sides Painted (5 Cubes): L1 Corners: The four cubes at the outermost corners of the base layer. With nothing above them, their top and two outer faces are exposed. Non-end L2: The one visible Level 2 piece that is not on an end (blocked on bottom and two sides). 4 Sides Painted (2 Cubes): L2 End Pieces: The two visible Level 2 cubes on the ends of the row. These are blocked only on the bottom and one lateral side. 5 Sides Painted (1 Cube): The Peak: The lone Level 3 cube at the very top. Only its bottom is blocked .

Question 15

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In the figure shown, how many cubes have exactly two sides painted?

  1. 3 cubes
  2. 4 cubes
  3. 5 cubes (correct answer)
  4. 6 cubes
  5. 7 cubes
Explanation: This structure consists of 20 total cubes. Following the standard DAT protocol, we assume all visible cubes are supported by columns beneath them and that every exposed surface is painted except for the bottom resting on the ground. Structural Breakdown Level 1 (Base): 15 cubes (arranged in a tiered 4×44 \times 4 pattern with the front corner omitted). Level 2: 4 cubes (forming a row that supports the higher tiers). Level 3: 1 cube (the topmost peak). Painted Sides Analysis To calculate the sides, we subtract the blocked faces (ground and neighboring cubes) from the total of 6 faces per cube. 1 Side Painted (7 Cubes):Interior L1s: The four interior base cubes have their tops blocked by Level 2 and their sides blocked by neighbors, leaving only their Top exposed (if not covered) or a single outer face. Sub-supports: The three Level 1 cubes that are directly beneath exactly one other block. 2 Sides Painted (5 Cubes): Tower Base & Mid: The Level 1 and Level 2 cubes at the back corner of the tower column.Middle Pieces: Three visible Level 1 cubes that are "middle" pieces (flanked by neighbors on both sides), leaving only the top and front faces exposed. 3 Sides Painted (5 Cubes): L1 Corners: The four cubes at the outermost corners of the base layer. With nothing above them, their top and two outer faces are exposed. Non-end L2: The one visible Level 2 piece that is not on an end (blocked on bottom and two sides). 4 Sides Painted (2 Cubes): L2 End Pieces: The two visible Level 2 cubes on the ends of the row. These are blocked only on the bottom and one lateral side. 5 Sides Painted (1 Cube): The Peak: The lone Level 3 cube at the very top. Only its bottom is blocked .

Question 16

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted

The figure shown is comprised of 20 cubes: the 13 explicitly pictured and the cubes necessary to support them. Of those 20 cubes, how many have exactly four sides painted?

  1. 1 cube
  2. 2 cubes
  3. 3 cubes (correct answer)
  4. 4 cubes
  5. 5 cubes
Explanation: This structure contains exactly 20 identical cubes arranged in a tiered V-shape with a four-story central tower. The analysis determines the number of painted sides for each cube by calculating its exposed surfaces, excluding the bottom surface resting on the ground. 0 Sides Painted (3 Cubes):These are the three "landlocked" support cubes at the base of the inner V. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and neighbors on all four lateral sides. 1 Side Painted (0 Cubes):In this specific structure, no cube has exactly one face exposed. Every cube either has zero exposure (if landlocked) or at least two faces peering out. 2 Sides Painted (8 Cubes)The Four-Story Tower Base: The bottom two cubes of the central tower are blocked on the top/bottom and two side neighbors. This leaves exactly two vertical faces exposed per cube .Middle Arm Supports (L1): The two cubes at the base of the Level 2 "middle arms" are blocked by ground, the cube above, and two lateral neighbors.Middle Arm Visible Blocks (L2): These two visible blocks are blocked on the bottom and two sides. With the top exposed, they have two faces painted (Top and Front).End Arm Supports (L1): The two cubes hidden beneath the Level 2 wingtips are blocked on the top, bottom, and two sides. This leaves the front and back faces exposed. 3 Sides Painted (5 Cubes)Bottom Row End Pieces: The four single-story cubes at the very ends of the base V have their top and two outer vertical faces exposed .Second Row Vertex: The cube at the vertex of the Level 2 V is blocked on the bottom and two lateral sides. This leaves the top, front, and back faces exposed. 4 Sides Painted (3 Cubes)Level 2 Wingtips: The two cubes at the far ends of the Level 2 V are blocked only on the bottom and one lateral side. This leaves the top and three vertical faces exposed.Third Level Tower Block: This cube is blocked on the top and bottom by other tower segments. All four of its vertical sides are exposed . 5 Sides Painted (1 Cube)The Topmost Cube: The cube at the very peak (Level 4) is blocked only on its bottom surface where it sits on Level 3. All five other faces are painted .

Question 17

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted

The figure shown is comprised of 20 cubes: the 13 explicitly pictured and the cubes necessary to support them. Of those 20 cubes, how many have exactly 1 side painted?

  1. No cubes (correct answer)
  2. 1 cube
  3. 2 cubes
  4. 3 cubes
  5. 4 cubes
Explanation: This structure contains exactly 20 identical cubes arranged in a tiered V-shape with a four-story central tower. The analysis determines the number of painted sides for each cube by calculating its exposed surfaces, excluding the bottom surface resting on the ground. 0 Sides Painted (3 Cubes):These are the three "landlocked" support cubes at the base of the inner V. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and neighbors on all four lateral sides. 1 Side Painted (0 Cubes):In this specific structure, no cube has exactly one face exposed. Every cube either has zero exposure (if landlocked) or at least two faces peering out. 2 Sides Painted (8 Cubes)The Four-Story Tower Base: The bottom two cubes of the central tower are blocked on the top/bottom and two side neighbors. This leaves exactly two vertical faces exposed per cube .Middle Arm Supports (L1): The two cubes at the base of the Level 2 "middle arms" are blocked by ground, the cube above, and two lateral neighbors.Middle Arm Visible Blocks (L2): These two visible blocks are blocked on the bottom and two sides. With the top exposed, they have two faces painted (Top and Front).End Arm Supports (L1): The two cubes hidden beneath the Level 2 wingtips are blocked on the top, bottom, and two sides. This leaves the front and back faces exposed. 3 Sides Painted (5 Cubes)Bottom Row End Pieces: The four single-story cubes at the very ends of the base V have their top and two outer vertical faces exposed .Second Row Vertex: The cube at the vertex of the Level 2 V is blocked on the bottom and two lateral sides. This leaves the top, front, and back faces exposed. 4 Sides Painted (3 Cubes)Level 2 Wingtips: The two cubes at the far ends of the Level 2 V are blocked only on the bottom and one lateral side. This leaves the top and three vertical faces exposed.Third Level Tower Block: This cube is blocked on the top and bottom by other tower segments. All four of its vertical sides are exposed . 5 Sides Painted (1 Cube)The Topmost Cube: The cube at the very peak (Level 4) is blocked only on its bottom surface where it sits on Level 3. All five other faces are painted .

Question 18

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

The figure shown is comprised of 20 cubes: the 14 explicitly pictured and the 6 cubes necessary to support them. Of those 20 cubes, how many have exactly five sides painted?

  1. No cubes
  2. 1 cube (correct answer)
  3. 2 cubes
  4. 3 cubes
  5. 4 cubes
Explanation: This structure consists of 20 total cubes (14 visible and 6 inferred as necessary supports). The side counts are determined by identifying how many faces of each cube are blocked by the ground (bottom) or neighboring cubes. Level 1 (Base Layer: 12 Cubes) 0 Sides Painted (3 Cubes):These are the three "landlocked" support bricks located directly beneath the visible Level 2 wings. They are blocked on all six faces: ground (bottom), the Level 2 cubes above (top), and horizontal neighbors on all four sides. 2 Sides Painted (3 Cubes): Front Middle Row: The two visible cubes in the front-middle of the base are blocked by ground, their neighbors, and the structure behind them, leaving the top and front faces exposed .Tower Base: The inferred Level 1 cube at the very base of the 4-stack tower. Due to the "jagged" formation, it has two faces exposed to the exterior. 3 Sides Painted (6 Cubes): Four Corner Pieces: These are the four outermost cubes of the base (far left/right and front corners). With nothing above them, their top and two vertical faces are exposed. Back Row Partial Supports: Two partially-shown Level 1 cubes in the back have three faces exposed. Level 2 (Second Layer: 4 Cubes) 2 Sides Painted (3 Cubes):Tower Support: One inferred Level 2 cube supporting the central tower.Wing Supports: The two visible Level 2 blocks that support the 4-stack tower. They are blocked on the top, bottom, and two sides. 3 Sides Painted (1 Cube):The single Level 2 block that has nothing resting atop it. It is blocked on the bottom and two sides, leaving the top, front, and back faces exposed. Level 3 (Third Layer: 3 Cubes) 2 Sides Painted (1 Cube):The inferred Level 3 cube that supports the peak of the 4-stack tower. 4 Sides Painted (2 Cubes):The two visible Level 3 cubes at the tops of the "steps." They are blocked only on the bottom and one lateral side by the central tower, leaving the top and three vertical faces exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube of the 4-stack tower is blocked only on its bottom surface. All five other faces (top and four sides) are exposed.

Question 19

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In Figure 1 as shown, how many cubes have exactly one of their exposed sides painted?

  1. 1 cube
  2. 2 cubes
  3. 3 cubes (correct answer)
  4. 4 cubes
  5. 5 cubes
Explanation: Figure 1 Explanation: Spatial Logic and Painted Sides To determine the number of painted sides, each cube must be analyzed by its position within the structure and the number of faces blocked by the ground (bottom) or neighboring cubes (top, front, back, left, or right) Level 1 (Base Layer: 8 Cubes)T he base layer is a 3×33 \times 3 grid with the front-left corner omitted. 1 Side Painted (3 Cubes):The Center Cube: Blocked by the ground and all four horizontal neighbors. Its top is the only exposed face.The Back-Middle and Middle-Left Cubes: These are blocked by the ground, three horizontal neighbors, and the Level 2 cubes resting directly on top of them. Only one outer vertical face is exposed for each. 2 Sides Painted (1 Cube):The Back-Left Corner: Blocked by the ground, the Level 2 corner cube above it, and two horizontal neighbors. Only the back and left vertical faces are exposed. 3 Sides Painted (4 Cubes):The Outer Corners and Edges: The remaining base cubes have their tops exposed along with two outer vertical faces. These include the back-right corner, the front-right corner, and the two edges bordering the "void" where the front-left corner was removed. Level 2 (Second Layer: 3 Cubes) These three cubes form an "L" shape directly above the back-left corner of the base. 2 Sides Painted (1 Cube):The Corner Cube: Blocked on the bottom (by Level 1), the top (by Level 3), and two horizontal sides (by its Level 2 "wings"). Only the back and left vertical faces are exposed. 4 Sides Painted (2 Cubes):The "Wings": These cubes are blocked only on the bottom and the one side connected to the corner. This leaves the top and three vertical faces exposed. Level 3 (Third Layer: 1 Cube) 4 Sides Painted (1 Cube):This single cube is blocked on the bottom (by Level 2) and the top (by Level 4). All four vertical faces are exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube is blocked only on its bottom side. The top and all four vertical faces are exposed.

Question 20

This figure has been made by cementing together cubes of the same size. After being cemented, the structure was painted on all sides EXCEPT FOR THE BOTTOM ON WHICH IT IS RESTING. The only hidden cubes are those required to support other cubes. You are to examine the figure closely to determine how many cubes have: only one of their sides painted only two of their sides painted only three of their sides painted only four of their sides painted all five of their sides painted Note: There are no cube-counting problems for which zero (0) is the correct answer.

In Figure 1 as shown, how many cubes have exactly three of their exposed sides painted?

  1. 1 cube
  2. 2 cubes
  3. 3 cubes
  4. 4 cubes (correct answer)
  5. 5 cubes
Explanation: Figure 1 Explanation: Spatial Logic and Painted Sides To determine the number of painted sides, each cube must be analyzed by its position within the structure and the number of faces blocked by the ground (bottom) or neighboring cubes (top, front, back, left, or right) Level 1 (Base Layer: 8 Cubes)T he base layer is a 3×33 \times 3 grid with the front-left corner omitted. 1 Side Painted (3 Cubes):The Center Cube: Blocked by the ground and all four horizontal neighbors. Its top is the only exposed face.The Back-Middle and Middle-Left Cubes: These are blocked by the ground, three horizontal neighbors, and the Level 2 cubes resting directly on top of them. Only one outer vertical face is exposed for each. 2 Sides Painted (1 Cube):The Back-Left Corner: Blocked by the ground, the Level 2 corner cube above it, and two horizontal neighbors. Only the back and left vertical faces are exposed. 3 Sides Painted (4 Cubes):The Outer Corners and Edges: The remaining base cubes have their tops exposed along with two outer vertical faces. These include the back-right corner, the front-right corner, and the two edges bordering the "void" where the front-left corner was removed. Level 2 (Second Layer: 3 Cubes) These three cubes form an "L" shape directly above the back-left corner of the base. 2 Sides Painted (1 Cube):The Corner Cube: Blocked on the bottom (by Level 1), the top (by Level 3), and two horizontal sides (by its Level 2 "wings"). Only the back and left vertical faces are exposed. 4 Sides Painted (2 Cubes):The "Wings": These cubes are blocked only on the bottom and the one side connected to the corner. This leaves the top and three vertical faces exposed. Level 3 (Third Layer: 1 Cube) 4 Sides Painted (1 Cube):This single cube is blocked on the bottom (by Level 2) and the top (by Level 4). All four vertical faces are exposed. Level 4 (Top Layer: 1 Cube) 5 Sides Painted (1 Cube):The topmost cube is blocked only on its bottom side. The top and all four vertical faces are exposed.