Painted Cube Questions Follow the Position of Each Small Cube

AI Summary

An AI-generated summary of this video lecture.

This lesson explains painted cube questions by linking each small cube's position to its number of painted faces. The instructor uses a table and 3D diagram to show that corner cubes have 3 painted faces, edge cubes (excluding corners) have 2, face-interior cubes have 1, and completely interior cubes have 0. Formulas are given: 8 corners; 12(n-2) edge cubes; 6(n-2)^2 face-interior cubes; and (n-2)^3 interior cubes. A useful result is derived: small cubes with at least two painted faces equal 8 + 12(n-2) = 12n - 16. The lesson then applies this to a word problem: an 18 cm cube painted on all faces is cut into 3 cm cubes. Since n = 18/3 = 6, the number of small cubes with exactly two painted faces is 12(6-2) = 48. The solution is verified by summing all categories: 8 + 48 + 96 + 64 = 216, which equals 6^3.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The video introduces painted cube questions with a table categorizing small cubes by position and number of painted faces. The instructor highlights the 'Painted faces' column, underlines 'On an edge, excluding corners', and circles the value 2 and formula 12(n - 2). A large cube diagram is annotated with red arrows pointing to edges and corners, visually connecting the table's abstract categories to specific locations on a 3D model. The full table is shown: corner (3 faces, 8), edge excluding corners (2 faces, 12(n-2)), inside a face excluding edges (1 face, 6(n-2)^2), and completely inside (0 faces, (n-2)^3).

2. 2:00 – 5:00 02:00-05:00

   The instructor progressively highlights each row of the table with red circles and boxes, moving from corner cubes (3 faces) through edge cubes (2 faces, formula 12(n-2)) to face-interior cubes (1 face, formula 6(n-2)^2) and completely interior cubes (0 faces). A 'Useful result' is derived: small cubes with at least two painted faces equal 8 + 12(n-2) = 12n - 16. A verification step shows that all four formulas sum to n^3, confirming the partition is complete. The segment transitions to a new slide titled 'Small Cubes with Exactly Two Painted Faces', presenting a word problem: an 18 cm cube painted yellow on all faces is cut into smaller cubes of side 3 cm each.

3. 5:00 – 7:04 05:00-07:04

   The instructor solves the word problem by first calculating n = 18/3 = 6, the number of small cubes along one edge. The formula for exactly two painted faces is applied: 12(n-2) = 12(6-2) = 48. A summary slide titled 'Two Painted Faces Means Edge Cubes' confirms the answer of 48. An independent verification sums all categories: 8 corners + 48 edge cubes + 96 face-interior cubes + 64 interior cubes = 216, which equals 6^3, confirming the solution is consistent with the total number of small cubes.

The core teaching progression moves from general classification to specific application. First, the instructor establishes a complete taxonomy of small cube positions within a larger divided cube, assigning each position a fixed number of painted faces and a counting formula. The key insight is that exactly two painted faces corresponds uniquely to edge cubes excluding corners, counted by 12(n-2). The 'at least two painted faces' result (12n - 16) combines corner and edge cubes. The worked example demonstrates the standard method: compute n from side lengths, identify the relevant position category, apply its formula, and verify by summing all categories to n^3. This verification step is a critical exam technique that catches arithmetic or categorization errors.