Geometrie & Funktionen · 3D Spatial Geometry (Surface Area & Volume), 3D Shapes & Nets, AMC 8 Spatial Reasoning.
Würfelfärbung & Körpernetz-Visualisierer
Färbung von n×n×n-Würfeln & Abwicklungen: Ecken, Kanten und Innenwürfel
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Experimentelle Steuerungsparameter
Modus
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mm
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Echtzeit-Herleitung & Kennzahlen
3 Flächen bemalt (Ecken)
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Always exactly 8 corners
2 Flächen bemalt (Kanten)
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12 edges × (n-2)
1 Fläche bemalt (Flächenmitte)
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6 faces × (n-2)²
0 Flächen bemalt (Kern)
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Unpainted inner core (n-2)³
Echtzeit-Formelherleitung
Grundformel:
--Einsetzungsschritte:
--Ergebnis: --
Grundschulversion:🎮 Quadernetz, Oberfläche und Volumen ↗
Typische Beispielaufgaben & Prüfungsfallen
5x5x5 Cube Complete Color Block Classification
Aufgabe: 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?
Analysestrategie: 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.
Schrittweise Herleitung:
- 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.
⚠️ Häufige Fehlvorstellung: Double-counting corner cubes on the edges; corners strictly belong to the 3-face painted category, so subtract 2 from each edge length.
Arbeitsblatt für den Unterricht & Hausaufgaben
Q1. A 4x4x4 painted cube is cut into 1x1x1 cubes. How many cubes have zero painted faces?
Answer Key: 8 (specifically (4-2)³ = 2³ = 8)
Q2. Which configuration can NEVER fold into a valid cube?
Answer Key: Nets containing a 2x2 square ('Tian' shape) or U-shaped concave indentations
Häufige Fragen & typische Missverständnisse
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!