1. Net → Cube Folding

What is this?

You are given a flat cube net and must imagine folding it into a cube.

The questions usually ask:

• Which faces become opposite?
• Which cube matches the given net?
• Which cube is impossible?

---

Easy Rules

✅ Squares sharing an edge in the net become adjacent.

✅ Opposite faces never touch.

✅ Three faces meeting at one corner must also meet on the folded cube.

---

Example

Cube Net

        E
        │
A ── B ── C ── D
            │
            F

---

Step 1: Fold the net

Imagine B as the front face.

        E
        │
A ── B ── C ── D
            │
            F

After folding,

• A folds to one side
• C folds to the other side
• E becomes the top
• F becomes the bottom
• D folds to the back

---

Step 2: Find opposite faces

A ↔ C

B ↔ D

E ↔ F

ASCII view

Opposite Faces

A <------> C

B <------> D

E <------> F

---

Step 3: Check a cube option

Suppose a cube shows

Top   : E
Front : A
Right : C

But A and C are opposite.

Opposite faces cannot touch.

❌ Impossible cube.

---

Final Answer

Opposite pairs

A ↔ C
B ↔ D
E ↔ F

Any cube showing A and C together is wrong.

---

2. Adjacent Face Rules

What is this?

Sometimes you don’t get a net.

Instead, you see the cube from different angles.

Your job is to find

• opposite faces
• adjacent faces

---

Important Rule

Whenever three faces appear together,

they meet at one corner.

Example

     Top
      X
     / \
    Y---Z

Since X, Y and Z are seen together,

they are all adjacent.

They can never be opposite.

---

Example

View 1

      X
     / \
    Y---Z

Visible together

X
Y
Z

So

X touches Y
X touches Z
Y touches Z

---

View 2

      X
     / \
    W---Z

Now X also touches

W

---

Step 1

Faces adjacent to X are

Y
Z
W

---

Step 2

Suppose the cube has

X
Y
Z
W
V

The only face never seen with X is

V

So

X is opposite V.

---

ASCII Table

Face X touches

✓ Y
✓ Z
✓ W

Never touches

✗ V

Therefore

X
↑
|
Opposite
|
V

---

Step 3

Can Y and Z be opposite?

They appear together in View 1.

      X
     / \
    Y---Z

Opposite faces never appear together.

So

Y and Z are adjacent.

---

Final Answer

X ↔ V

Y and Z are adjacent.

---

Quick Tricks for Exams

From a Net

Touching squares
      ↓
Adjacent faces

Never touching after folding
      ↓
Opposite faces

---

From Multiple Views

Appears with X
      ↓
Adjacent to X

Never appears with X
      ↓
Opposite to X

---

Golden Rule

Opposite faces

❌ Never touch
❌ Never appear together
❌ Never share an edge

Adjacent faces

✔ Touch each other
✔ Can appear in the same view
✔ Meet at one corner
Here are simple notes for the remaining **Cube** topics in the same style.

---

# 3. Hidden Face & Cube Rotation

## What is this?

A cube is shown from different angles after being rotated.

You need to find:

* the hidden face
* which face comes to the top/front/right after rotation
* whether two pictures show the same cube

---

## Easy Rules

✅ Rotating a cube **does not change** opposite faces.

✅ Adjacent faces always remain adjacent.

✅ Only the cube's **position changes**, not the numbering.

---

## Example

Suppose a cube has

```text
        1
      ┌───┐
   2  │   │ 3
      └───┘
        6

Back = 4

Opposite faces are

1 ↔ 6

2 ↔ 3

4 ↔ 5

---

Initial Position

Top   = 1

Front = 2

Right = 4

---

Rotate the cube

Suppose the cube is rotated so that

Front becomes Top

Visualize it like tilting the cube towards you.

Before

      Top
       1

Front 2

After

      Top
       2

Front 6

The old top moves to the back.

---

Hidden Face Example

A cube shows

Top   = 1

Front = 2

Right = 4

Which face is hidden on the bottom?

Use opposite faces.

1 ↔ 6

So

Bottom = 6

---

Quick Trick

Visible faces
─────────────
Top
Front
Right

Hidden faces
─────────────
Bottom
Back
Left

If you know the opposite pairs, you can immediately find every hidden face.

---

Final Answer

Rotation changes positions.

Opposite faces NEVER change.

---

4. Painted Cube Cutting Problems

What is this?

A large cube is painted and then cut into many small cubes.

Questions ask:

• How many cubes have 3 painted faces?
• 2 painted faces?
• 1 painted face?
• No painted faces?

---

Example

A cube is painted on all six sides.

It is cut into

3 × 3 × 3

small cubes.

---

Step 1

Imagine the cube

        _______
       /______/|
      /______/ |
      |      | |
      |      | /
      |______|/

Every outside face is painted.

---

Step 2

After cutting

3 × 3 × 3

= 27 cubes

---

Corner Cubes

Corners touch

Top
Front
Side

So each corner has

3 painted faces

There are always

8 corners

Answer = 8

---

Edge Cubes

Edge cubes lie here

Corner ●────● Corner
          □

The middle cube touches

• two painted faces

Formula

12 × (n−2)

For n = 3

12 × 1

= 12

---

Face Cubes

These are inside each face.

□□□□□
□ ■ □
□□□□□

The center cube has only

1 painted face

Formula

6 × (n−2)²

For n = 3

6 × 1²

= 6

---

Inner Cubes

These never touch the paint.

Formula

(n−2)³

For n = 3

1³

= 1

---

Final Table

Painted Faces	Formula
3	8
2	12(n−2)
1	6(n−2)²
0	(n−2)³

---

Example

For

4 × 4 × 4

3 faces = 8

2 faces = 12×2 = 24

1 face = 6×4 = 24

0 faces = 8

---

Quick Trick

Always remember

Corners
= 8

Edges
= 12

Faces
= 6

---

5. Cube Cutting (Maximum Pieces)

What is this?

A cube is cut several times.

Find the maximum number of pieces possible.

---

Easy Rule

Every new cut should pass through as many existing pieces as possible.

This gives the maximum number of pieces.

---

Formula

For k cuts

Maximum Pieces

= (k³ + 5k + 6) / 6

---

Example 1

One cut

Cube

↓

│

Pieces

2

Formula

(1³+5+6)/6

=12/6

=2

---

Example 2

Two cuts

│

──

Maximum

4 pieces

Formula

(8+10+6)/6

=24/6

=4

---

Example 3

Three cuts

Formula

(27+15+6)/6

=48/6

=8

---

Example 4

Four cuts

(64+20+6)

=90

90/6

=15 pieces

---

Quick Table

Cuts	Maximum Pieces
0	1
1	2
2	4
3	8
4	15
5	26
6	42

---

Exam Trick

If the question is “maximum number of pieces”, directly use the formula.

If the question is about painted cubes, use the painted cube formulas instead.

---

Quick Revision

Hidden Face
───────────
Use opposite faces.

Rotation never changes opposite faces.

Painted Cube
────────────
3 faces = 8

2 faces = 12(n−2)

1 face = 6(n−2)²

0 faces = (n−2)³

Maximum Pieces
──────────────
(k³ + 5k + 6)
/────────────
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