Question 14

Step 1 · Analyze Side and Top Views of the 'C' Shape

The initial $8$ cubes have coordinates $(x, y, z)$:

• Layer 1 (front, $z = 0$): $(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 2, 0), (1, 2, 0)$
• Layer 2 (back, $z = 1$): $(0, 0, 1), (0, 1, 1), (0, 2, 1)$

Front view (projection on $x\text{-}y$ plane):

Side view (projection on $y\text{-}z$ plane): Collecting all $(y, z)$ pairs: $(0, 0), (1, 0), (2, 0) \quad \text{and} \quad (0, 1), (1, 1), (2, 1)$

This forms a rectangle of dimensions $3 \times 2$.

Top view (projection on $x\text{-}z$ plane): Collecting all $(x, z)$ pairs: $(0, 0), (1, 0) \quad \text{and} \quad (0, 1)$

This forms an 'L' shape.

Step 1 · Construct Shape with 'C' Front and 'A' Top Views

Extending to a $3 \times 3 \times 3$ grid, let the coordinate sets be:

• Front view ('C'): $C_{xy} = \{(0, 0), (1, 0), (2, 0), (0, 1), (0, 2), (1, 2), (2, 2)\}$
• Top view ('A'): $A_{xz} = \{(0, 0), (1, 0), (2, 0), (0, 1), (2, 1), (0, 2), (2, 2)\}$

A cube exists at $(x, y, z)$ if and only if $(x, y) \in C_{xy}$ and $(x, z) \in A_{xz}$:

• For $x = 0$:

  • $(0, 0) \in C_{xy} \implies z \in \{0, 1, 2\} \rightarrow 3 \text{ cubes}$
  • $(0, 1) \in C_{xy} \implies z \in \{0, 1, 2\} \rightarrow 3 \text{ cubes}$
  • $(0, 2) \in C_{xy} \implies z \in \{0, 1, 2\} \rightarrow 3 \text{ cubes}$
  • Total for $x = 0$: $3 + 3 + 3 = 9 \text{ cubes}$

• For $x = 1$:

  • $(1, 0) \in C_{xy} \implies z = 0 \rightarrow 1 \text{ cube}$
  • $(1, 1) \notin C_{xy} \rightarrow 0 \text{ cubes}$
  • $(1, 2) \in C_{xy} \implies z = 0 \rightarrow 1 \text{ cube}$
  • Total for $x = 1$: $1 + 0 + 1 = 2 \text{ cubes}$

• For $x = 2$:

  • $(2, 0) \in C_{xy} \implies z \in \{0, 1, 2\} \rightarrow 3 \text{ cubes}$
  • $(2, 1) \in C_{xy} \implies z \in \{0, 1, 2\} \rightarrow 3 \text{ cubes}$
  • $(2, 2) \in C_{xy} \implies z \in \{0, 1, 2\} \rightarrow 3 \text{ cubes}$
  • Total for $x = 2$: $3 + 3 + 3 = 9 \text{ cubes}$

$\text{Total cubes} = 9 + 2 + 9 = 20$

Step 1 · Construct Shape with 'C' Front, 'A' Top, and 'F' Side Views

In addition to $C_{xy}$ and $A_{xz}$, the $3 \times 3$ side view 'F' is: $F_{yz} = \{(0, 0), (1, 0), (2, 0), (0, 1), (2, 1), (0, 2), (1, 2), (2, 2)\}$ (All pairs in $3 \times 3$ except $(1, 1)$).

A cube exists at $(x, y, z)$ if $(x, y) \in C_{xy}$, $(x, z) \in A_{xz}$, and $(y, z) \in F_{yz}$:

• For $x = 0$:

  • $y = 0 \implies z \in \{0, 1, 2\} \rightarrow (0, 0), (0, 1), (0, 2) \in F_{yz} \rightarrow 3 \text{ cubes}$
  • $y = 1 \implies z \in \{0, 2\} \rightarrow (1, 0), (1, 2) \in F_{yz} \text{ (since } (1, 1) \notin F_{yz}\text{)} \rightarrow 2 \text{ cubes}$
  • $y = 2 \implies z \in \{0, 1, 2\} \rightarrow (2, 0), (2, 1), (2, 2) \in F_{yz} \rightarrow 3 \text{ cubes}$
  • Total for $x = 0$: $3 + 2 + 3 = 8 \text{ cubes}$

• For $x = 1$:

  • $y = 0 \implies z = 0 \rightarrow (0, 0) \in F_{yz} \rightarrow 1 \text{ cube}$
  • $y = 1 \implies (1, 1) \notin C_{xy} \rightarrow 0 \text{ cubes}$
  • $y = 2 \implies z = 0 \rightarrow (2, 0) \in F_{yz} \rightarrow 1 \text{ cube}$
  • Total for $x = 1$: $1 + 0 + 1 = 2 \text{ cubes}$

• For $x = 2$:

  • $y = 0 \implies z \in \{0, 1, 2\} \rightarrow 3 \text{ cubes}$
  • $y = 1 \implies z \in \{0, 2\} \rightarrow 2 \text{ cubes}$
  • $y = 2 \implies z \in \{0, 1, 2\} \rightarrow 3 \text{ cubes}$
  • Total for $x = 2$: $3 + 2 + 3 = 8 \text{ cubes}$

$\text{Total cubes} = 8 + 2 + 8 = 18$

Step 1 · Construct a Solid with All 'L' Views

Consider a $2 \times 2 \times 2$ solid whose front, top, and side views are all 'L' shapes:

• Front view: $L_{xy} = \{(0, 0), (1, 0), (0, 1)\}$
• Top view: $L_{xz} = \{(0, 0), (1, 0), (0, 1)\}$
• Side view: $L_{yz} = \{(0, 0), (1, 0), (0, 1)\}$

A cube $(x, y, z)$ exists if $(x, y) \in L_{xy}$, $(x, z) \in L_{xz}$, and $(y, z) \in L_{yz}$:

• $(0, 0, 0)$: Valid $\rightarrow 1 \text{ cube}$
• $(1, 0, 0)$: Valid $\rightarrow 1 \text{ cube}$
• $(0, 1, 0)$: Valid $\rightarrow 1 \text{ cube}$
• $(0, 0, 1)$: Valid $\rightarrow 1 \text{ cube}$

All other coordinates fail at least one view condition. $\text{Total cubes} = 4$