Analyze 3D Cubes - DAT Perceptual Ability

1. What formula gives exposed faces for one cube with $k$ glued faces (shared faces)?

   $6-k$. Each glued face reduces the exposed count by one from the initial six surfaces of a cube.

2. What is the exposed face count for a solid $4\times 3\times 2$ rectangular prism of unit cubes?

   52. Prism formula $2(ab + ac + bc)$ with $a=4$, $b=3$, $c=2$ computes the total surface area.

3. What is the exposed face count for a solid $3\times 3\times 2$ rectangular prism of cubes?

   42. Applying $2(ab + ac + bc)$ to $a=3$, $b=3$, $c=2$ sums the exposed areas of each dimension pair.

4. What is the number of exposed faces on a cube with exactly 3 faces glued to other cubes?

   3. Three glued faces each share a surface with adjacent cubes, resulting in three remaining visible faces.

5. What is the exposed face count for a solid $3\times 2\times 1$ rectangular prism of cubes?

   22. Prism formula $2(ab + ac + bc)$ with $a=3$, $b=2$, $c=1$ calculates paired face areas.

6. What is the exposed face count of a solid $a\times b\times c$ rectangular prism of unit cubes?

   $2(ab+ac+bc)$. Pairs of opposite faces contribute areas $ab$, $ac$, and $bc$, each doubled for both sides of the prism.

7. What is the exposed face count for a $2\times 2\times 1$ block of cubes?

   16. Using the slab formula $2ab + 2a + 2b$ with $a=2$, $b=2$ yields the total exposed surfaces.

8. What is the exposed face count for a solid $3\times 2\times 2$ rectangular prism of cubes?

   32. Using $2(ab + ac + bc)$ for $a=3$, $b=2$, $c=2$ accounts for all external surfaces.

9. What is the exposed face count for a straight line of $n$ cubes (a $1\times 1\times n$ rod)?

   $4n+2$. End cubes expose five faces each ($10$ total), while interior cubes expose four, summing appropriately for length $n$.

10. What is the exposed face count for a solid $2\times 2\times 2$ cube of unit cubes?

    24. The cube formula $6n^2$ for $n=2$ represents the total surface area in unit faces.

11. What is the exposed face count for a $4\times 1\times 1$ line of cubes?

    18. Applying $4n + 2$ for $n=4$ captures the increased exposures from additional interior cubes in the chain.

12. What is the number of exposed faces on a cube with exactly 5 faces glued to other cubes?

    1. Five glued faces hide all but one surface, as only the unattached face remains exposed.

13. What is the exposed face count for a $2\times 3\times 1$ slab of cubes?

    22. The slab formula $2ab + 2a + 2b$ with $a=2$, $b=3$ sums top, bottom, and side exposures.

14. What is the number of exposed faces on a cube with exactly 1 face glued to another cube?

    5. Gluing one face to another cube obscures that single face, reducing exposed surfaces from six to five.

15. What is the maximum number of exposed faces on a single isolated cube?

    6. An isolated cube has no faces obscured by attachments, leaving all six surfaces visible.

16. Identify the exposed faces for two cubes glued together on one face (a $1\times 1\times 2$ rod).

    10. Each cube starts with six faces, but the shared face hides two (one per cube), totaling ten exposed.

17. What is the number of exposed faces on a cube with exactly 2 faces glued to other cubes?

    4. Attaching two faces to other cubes hides those two surfaces, leaving four exposed on the original cube.

18. What is the exposed face count for a $3\times 1\times 1$ line of cubes?

    14. The rod formula $4n + 2$ applied to $n=3$ accounts for end and interior exposures in the line.

19. What is the total exposed faces if $N=5$ cubes form a chain with $A=4$ adjacencies?

    $22$. Formula $6N - 2A$ with $N=5$, $A=4$ subtracts hidden faces from total possible exposures.

20. What is the surface area (exposed faces) of a solid $n\times n\times n$ cube of unit cubes?

    $6n^2$. The surface area consists of six faces, each comprising $n^2$ unit squares, totaling $6n^2$ exposed unit faces.

21. What is the total exposed faces if $N=6$ cubes have $A=7$ adjacent touching pairs?

    $22$. Using $6N - 2A$ for $N=6$, $A=7$ deducts twice the adjacencies from overall cube faces.

22. What is the exposed face count for a flat $a\times b\times 1$ rectangular slab of cubes?

    $2ab+2a+2b$. Top and bottom layers fully expose $2ab$ faces, with perimeter sides adding $2a+2b$ exposed faces.

23. What is the number of exposed faces on a cube with exactly 4 faces glued to other cubes?

    2. Four attached faces obscure those surfaces, yielding two visible faces on the cube.

24. What is the total exposed faces formula using $N$ cubes and $A$ adjacent touching pairs?

    $6N-2A$. Starting with $6N$ total faces, each adjacency hides two faces, one from each connected cube.

25. What is the number of exposed faces on a cube completely surrounded (all 6 faces glued)?

    0. Complete enclosure by other cubes on all sides obscures every face, leaving none visible.

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