Spatial Geometry & Functions · 3D Spatial Geometry (Surface Area & Volume), 3D Shapes & Nets, AMC 8 Spatial Reasoning.

Painted Cube & Polyhedron Net Visualizer

Spatial reasoning breakthrough: 3D exploded view revealing why vertex corners are constant 8, edge centers are 12(n-2), face centers are 6(n-2)², and the unpainted core is (n-2)³. Test all 11 valid cube nets with real-time fold animations.

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Live Derivations & Metrics

count_3face

Always exactly 8 corners

count_2face

12 edges × (n-2)

count_1face

6 faces × (n-2)²

count_0face

Unpainted inner core (n-2)³

Real-Time Formula Derivation

Governing Formula: --

Substitution: --

Regime: --

Worked Examples & Exam Question Deep Dive

5x5x5 Cube Complete Color Block Classification

Problem: A 5×5×5 cube painted red on all sides is sliced into 125 small 1×1×1 unit cubes. How many unit cubes have exactly 2 red faces and exactly 1 red face?

Analytical Strategy: 2-face painted blocks sit on the 12 edges excluding corners: 12 × (5-2) = 36. 1-face painted blocks sit inside the 6 outer faces: 6 × (5-2)² = 54.

Step-by-Step Derivation:

Step 1: Total unit cubes N = 5³ = 125, division parameter n = 5.
Step 2: 3-face cubes located at 8 vertices: constant 8.
Step 3: 2-face cubes on 12 edges: 12 × (5 - 2) = 36.
Step 4: 1-face cubes in 6 face centers: 6 × (5 - 2)² = 6 × 9 = 54.
Step 5: 0-face unpainted core cubes: (5 - 2)³ = 27.
Step 6: Verification check: 8 + 36 + 54 + 27 = 125.

⚠️ Common Misconception Pitfall: Double-counting corner cubes on the edges; corners strictly belong to the 3-face painted category, so subtract 2 from each edge length.

Student Inquiry Worksheet & Practice Problems

Q1. A 4x4x4 painted cube is cut into 1x1x1 cubes. How many cubes have zero painted faces?

Q2. Which configuration can NEVER fold into a valid cube?

Frequently Asked Questions & Misconceptions

What is the mnemonic rule for identifying the 11 valid cube nets?

Mnemonic: 6 variations of 1-4-1, 3 of 2-3-1, 1 of 2-2-2 (staircase), 1 of 3-3; any 2x2 block or concave U-pocket immediately fails!