Count Hidden Blocks - Air Force Officer Qualifying Test AFOQT

1. What does the term full-face contact mean for unit cubes in a grid stack?

   Two cubes share a complete square face (same area and alignment). Full-face contact requires complete overlap of faces, ensuring the shared area is fully covered.

2. What is the rule to decide whether a side face is visible in a standard isometric drawing (view from above)?

   A side face is visible if it is on the outer boundary in that direction. In isometric views, side faces are visible only if no other blocks are positioned outward in that direction from the cube.

3. What is the number of exposed faces of a cube that has exactly three full-face neighbors on three different faces?

   3 exposed faces. Three full-face neighbors on different faces cover three faces, leaving three exposed.

4. What is the maximum number of exposed faces a single isolated cube can have?

   6 exposed faces. Without any contacts or occlusions, all six faces of the cube are part of the external surface.

5. What is the exposed-face total for $n$ isolated cubes with no contacts between any cubes?

   $6n$. Each isolated cube contributes six exposed faces, with no subtractions for contacts.

6. Find the exposed-face total for three cubes stacked vertically in a single column.

   $14$. Vertical stacking with two contacts yields $6\times^3 - 2\times^2 = 14$ exposed faces.

7. Find the exposed-face total for two cubes that share exactly one full face.

   $10$. With one contact, the total exposed faces are $6\times^2 - 2\times^1 = 10$.

8. Identify the internal faces created when two cubes are stacked directly (perfect alignment).

   The shared interface: bottom of top cube and top of bottom cube. Perfect alignment in stacking internalizes the interfacing faces, hiding them from exposure.

9. What is the quickest counting method for exposed faces using adjacency information?

   Start with $6n$ and subtract $2$ for each full-face contact between cubes. The formula accounts for each cube starting with six faces and each contact hiding two faces.

10. What is the definition of a hidden surface in a stacked-block figure from a given viewpoint?

    A face not visible because another block occludes it from that viewpoint. Occlusion occurs when a block is positioned in the line of sight, making the face behind it invisible from the given perspective.

11. What happens to a cube face that is in full contact with another cube face (shared face)?

    It is completely hidden (an internal face) and never visible. Full contact with another face creates an internal interface that is not part of the external surface.

12. What is the hidden-surface rule when cubes touch only along an edge or a corner?

    No face is hidden by contact; only viewpoint occlusion can hide faces. Edge or corner contacts do not cover entire faces, leaving all faces exposed unless hidden by viewpoint occlusion.

13. Find the exposed-face total for four cubes in a straight line with full-face contacts.

    $18$. Three contacts in the line give $6\times^4 - 2\times^3 = 18$.

14. What is the number of exposed faces of a cube that has exactly two full-face neighbors on different faces?

    4 exposed faces. Two full-face neighbors on different faces cover two faces, leaving four exposed.

15. Find the number of hidden faces created by $k$ full-face contacts between unit cubes.

    $2k$ hidden faces. Each full-face contact hides exactly two faces, one from each cube.

16. Which faces are candidates to be hidden by vertical stacking (one cube directly on top of another)?

    The lower cube's top face and the upper cube's bottom face. Vertical stacking directly covers the top face of the lower cube and the bottom face of the upper cube.

17. A structure has $n$ cubes and $k$ full-face contacts. What is the total number of exposed faces?

    $6n-2k$. The formula derives from each cube's six faces minus two per contact for the hidden pairs.

18. What is the hidden-surface rule for two cubes that touch along an entire face?

    Both touching faces are hidden; they do not count as exposed surfaces. Each full-face contact hides one face from each cube, removing them from the count of exposed surfaces.

19. Find the exposed-face total for a $2\times 1\times 1$ rectangular prism made of unit cubes.

    $10$. The $2\times^1\times^1$ has one contact, so $6\times^2 - 2 = 10$.

20. Find the exposed-face total for a $2\times 2\times 1$ slab made of unit cubes.

    $16$. The slab has four contacts, resulting in $6\times^4 - 8 = 16$.

21. What is the definition of an exposed surface in a stacked-block figure from a given viewpoint?

    A face visible from the viewpoint and not covered by any adjacent block. Exposure requires no blocks blocking the line of sight, allowing the face to be seen in the figure.

22. Find the exposed-face total for a solid $2\times 2\times 2$ cube of unit cubes.

    $24$. The $2\times^2\times^2$ has 12 contacts, leading to $48 - 24 = 24$ exposed faces.

23. Find the exposed-face total for three cubes in a straight line (each middle contact is full-face).

    $14$. With two contacts, the total is $6\times^3 - 2\times^2 = 14$.

24. What is the number of exposed faces of a cube that has exactly one full-face neighbor?

    5 exposed faces. One full-face neighbor covers one face, reducing the exposed count from six to five.

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