Painted Cube Cutting Calculator
Enter the number of cuts/divisions per side (N) to find the breakdown of smaller cubes by painted faces.
Results Breakdown
Step-by-Step Solution:
How to Solve Painted Cube Problems (Aptitude & Logic)
1. 3-Side Painted Cubes Formula
For any fully painted cube cut into N × N × N smaller cubes, 3-side painted cubes are always located at the 8 corners. Answer is always 8 (for N ≥ 2).
2. 2-Side Painted Cubes Formula
These lie along the 12 edges of the cube (excluding corners): 12 × (N - 2).
3. 1-Side Painted Cubes Formula
These lie on the 6 flat faces: 6 × (N - 2)².
4. 0-Side Painted Cubes Formula
These are inner unpainted cubes: (N - 2)³.
Painted Cube Cutting Calculator & Step-by-Step Problem Solver
Understanding Painted Cube Cutting Problems: The Complete Guide
Painted cube cutting problems are among the most popular logical reasoning and spatial visualization questions asked in competitive entrance exams, including GRE, GMAT, SAT, UPSC, SSC, and corporate job placement tests. While these problems look complicated at first glance, they follow a predictable mathematical pattern based on three-dimensional geometry.
This comprehensive guide explains how a painted cube behaves when cut into smaller identical cubes, the mathematical derivations behind each formula, practical examples, and shortcuts to solve these questions instantly without manual counting.
What is a Painted Cube Cutting Problem?
In a standard painted cube scenario, a large solid cube is first coated with paint on all six of its outer faces. The large cube is then sliced evenly along its length, width, and height into smaller identical unit cubes. The objective is to determine how many smaller cubes have 3 painted faces, 2 painted faces, 1 painted face, or no painted faces (unpainted or inner cubes).
If a large cube with side length L is divided into smaller unit cubes with side length l, the number of divisions or cuts per edge (N) is given by:
N = L / l
The total number of smaller unit cubes generated after slicing is always N³.
Mathematical Formulas & Logical Derivations
To master these aptitude questions, you do not need to memorize random formulas. Understanding where each type of cube is located inside the original 3D shape makes solving any variation simple.
Summary Table of Cube Formulas
| Cube Classification | Physical Location on Big Cube | Mathematical Formula |
|---|---|---|
| 3-Side Painted Cubes | Located at the 8 outer corners | 8 (Always constant for N ≥ 2) |
| 2-Side Painted Cubes | Located along the 12 outer edges (excluding corners) | 12 × (N - 2) |
| 1-Side Painted Cubes | Located on the surfaces of the 6 flat faces | 6 × (N - 2)² |
| 0-Side Painted (Unpainted) | Located deep inside the core (inner core cube) | (N - 2)³ |
1. Three Painted Faces (Corner Cubes)
Corner cubes sit at the 8 vertices of the large cube. Because three plane surfaces meet at every corner of a cube, these corner unit cubes expose 3 of their faces to the outside environment. As long as N ≥ 2, every painted cube will always have exactly 8 cubes with 3 painted faces, regardless of how large N becomes.
2. Two Painted Faces (Edge Cubes)
Edge cubes lie along the 12 straight borders connecting the corners. The unit cubes located at the extreme ends of an edge are corner cubes (which have 3 painted sides). Therefore, along any single edge of length N, the number of unit cubes with exactly 2 painted faces is (N - 2). Multiplying this by the total 12 edges gives the universal formula: 12 × (N - 2).
3. One Painted Face (Central Face Cubes)
These unit cubes sit in the middle of each flat surface (face) of the big cube. Since a cube has 6 identical square faces, we calculate the area of the inner square formed by removing the boundary rows and columns. For each face, the number of 1-side painted cubes is (N - 2)². Multiplying by 6 faces yields: 6 × (N - 2)².
4. Zero Painted Faces (Inner Unpainted Cubes)
The core cubes lie deep inside the big cube and receive no paint whatsoever. Removing a 1-unit layer from all outer faces leaves an inner block of unpainted cubes with dimensions (N - 2) × (N - 2) × (N - 2). The total number of unpainted cubes is calculated using (N - 2)³.
Step-by-Step Practical Example Solution
Question: A solid wooden cube painted red on all sides is cut into smaller cubes such that N = 5. Find the breakdown of all smaller cubes generated.
Step-by-Step Solution:
Total Cubes generated: N³ = 5³ = 125 total cubes.
3-Side Painted Cubes: Located at the corners = 8 cubes.
2-Side Painted Cubes: 12 × (5 - 2) = 12 × 3 = 36 cubes.
1-Side Painted Cubes: 6 × (5 - 2)² = 6 × 3² = 6 × 9 = 54 cubes.
0-Side Painted Cubes: (5 - 2)³ = 3³ = 27 cubes.
Verification Check: 8 + 36 + 54 + 27 = 125 total cubes. (The sum matches perfectly).
Frequently Asked Questions About Painted Cube Problems
Q1. What is a Painted Cube Problem?
A Painted Cube Problem is a common mathematics and reasoning question in which a large cube is painted on all six sides and then cut into smaller equal-sized cubes. The task is to determine how many small cubes have 3, 2, 1, or 0 painted faces based on the size of the original cube.
Q2. How many cubes have 3 faces painted?
For a cube divided into smaller cubes, there are always 8 corner cubes with 3 faces painted, as long as N ≥ 2. These cubes are located at the eight corners of the original cube.
Q3. What is the formula for cubes with 2 faces painted?
The formula for finding the number of small cubes with exactly 2 painted faces is:
12 × (N − 2)
These cubes are located along the edges of the large cube, excluding the corner cubes.
Q4. How many unpainted cubes are there in a 5×5×5 cube?
For a 5×5×5 cube, the number of completely unpainted cubes is calculated using:
(N − 2)³
Therefore:
(5 − 2)³ = 3³ = 27
So, there are 27 completely unpainted cubes.
Q5. Can the same formulas be used for a cuboid?
No. A cuboid has three potentially different dimensions, so the standard formulas for a cube cannot be applied directly. Cuboid problems require formulas based on the length, width, and height of the cuboid.
Q6. Why is a Painted Cube Calculator useful for exams?
A Painted Cube Calculator can help students quickly check their answers and understand the concept behind painted cube questions. You can enter different values of N to practice various problems, verify calculations, and improve your speed and accuracy for competitive exams.
Q7. What if only some faces of the cube are painted?
If only certain faces of the cube are painted, the standard formulas for a fully painted cube may not apply. In such cases, you need to carefully analyze which faces are painted and determine the number of cubes affected by the painted surfaces.
Q8. How do you calculate the total number of painted cubes?
The total number of small cubes with at least one painted face can be calculated by subtracting the completely unpainted cubes from the total number of small cubes.
Total Painted Cubes = Total Cubes − Unpainted Cubes
Therefore:
Total Painted Cubes = N³ − (N − 2)³
Conclusion
The Painted Cube Cutting Calculator is a useful tool for understanding and practicing painted cube problems in mathematics and competitive exams. By learning the basic formulas and testing different values of N, students can quickly verify their calculations and strengthen their understanding of cube-cutting concepts.
Regular practice with different cube sizes can help improve calculation speed, accuracy, and problem-solving skills. Use the calculator to check your answers, practice different examples, and become more confident with painted cube questions.
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