Question 7

A net of a cube is a two-dimensional shape that can be folded along its edges to form a three-dimensional cube without any overlaps or gaps. A cube has 6 faces, so a net must consist of exactly 6 squares. We can determine if a figure is a net of a cube by visualizing its folding process.

Step 1 — Analyze Figure (i)

Let's label the squares in Figure (i) for clarity: A B C D E F

We choose square D as the base of the cube. The faces directly connected to D (sharing an edge with D) are C, E, and B. These will form three of the side faces of the cube. The remaining two squares are A and F. Square A is connected to B, which is connected to D. This means A is adjacent to D. Square F is connected to E, which is connected to D. This means F is also adjacent to D. If D is the base, then the face opposite to it (the top face) must not be adjacent to D. However, both A and F are adjacent to D. This means neither A nor F can be the top face. Alternatively, if we try to fold it: If D is the base, C is the left face, E is the right face, and B is the back face. F is attached to E, so it folds up from the right face to become the front face. A is attached to B, so it folds up from the back face to become the top face. Now, consider the top-front-left corner of the cube. The top edge of the left face (C) must meet the left edge of the top face (A). The right edge of the front face (F) must meet the front edge of the top face (A). However, if we trace the path from C to F in the net: C-D-E-F. These two faces are separated by two other faces (D and E). When folded, they would not meet at the same corner. There would be a gap or an overlap. Therefore, Figure (i) cannot form a cube.

(i) No

Step 2 — Analyze Figure (ii)

Let's label the squares in Figure (ii): A B C D E F Here, E is attached to B, and F is attached to C.

Let's try to choose a base. If we choose B as the base: A folds up as the left face. C folds up as the right face. E folds up as the back face. D is attached to C, so D folds up from the right face to become the front face. F is attached to C, so F folds up from the right face to become the top face. This means that both D and F would try to occupy the front/top positions relative to the base, leading to an overlap. If we choose C as the base: B folds up as the left face. D folds up as the right face. F folds up as the back face. A is attached to B, so A folds up from the left face to become the front face. E is attached to B, so E folds up from the left face to become the top face. This means that both A and E would try to occupy the front/top positions relative to the base, leading to an overlap. Since no square can be chosen as a base without causing overlaps, Figure (ii) cannot form a cube.

(ii) No

Step 3 — Analyze Figure (iii)

Let's label the squares in Figure (iii): A B C D E F

Let's choose C as the base. B folds up as the left face. D folds up as the right face. A folds up from C as the back face. E is attached to B, so E folds up from the left face to become the front face. F is attached to D, so F folds up from the right face to become the top face. When folded, all faces are distinct and meet correctly without overlaps or gaps. Base: C Left: B Right: D Back: A Front: E Top: F This arrangement forms a cube.

(iii) Yes

Step 4 — Analyze Figure (iv)

Let's label the squares in Figure (iv): A B C D E F

This is the classic "cross" net. Let's choose C as the base. B folds up as the left face. D folds up as the right face. A folds up as the back face. E folds up as the front face. F is attached to E, so F folds up from the front face to become the top face. When folded, all faces are distinct and meet correctly without overlaps or gaps. Base: C Left: B Right: D Back: A Front: E Top: F This arrangement forms a cube.

(iv) Yes

Step 5 — Analyze Figure (v)

Let's label the squares in Figure (v): A B C D E F Here, E is attached to C, and F is attached to E.

Let's choose C as the base. B folds up as the left face. D folds up as the right face. E folds up as the back face. A is attached to B, so A folds up from the left face to become the front face. F is attached to E, so F folds up from the back face to become the top face. When folded, all faces are distinct and meet correctly without overlaps or gaps. Base: C Left: B Right: D Back: E Front: A Top: F This arrangement forms a cube.

(v) Yes

Step 6 — Analyze Figure (vi)

Let's label the squares in Figure (vi): A B C D E F

Let's choose B as the base. A folds up as the left face. C folds up as the right face. E is attached to A, so E folds up from the left face to become the back face. D folds up as the front face. F is attached to D, so F folds up from the front face to become the top face. When folded, all faces are distinct and meet correctly without overlaps or gaps. Base: B Left: A Right: C Back: E Front: D Top: F This arrangement forms a cube.

(vi) Yes

Answer

(i) No (ii) No (iii) Yes (iv) Yes (v) Yes (vi) Yes