Grades 9-10 Video Solutions 2024-2024_9-10

Video Transcription

RL has n cubed, where n greater than 2, identical small cubes. Use these to make a large cube and paint the entire outer surface of the large cube.  The number of small cubes with only one face painted is equal to the number of those with no face painted. What is the value of n?  There are n minus 2 squared small cubes on each face of the large cube that have a single painted face.  And therefore we have 6 times n minus 2 squared small cubes with a single face painted. Note also that there are n minus 2 cubed unpainted cubes on the interior of the big cube.  And thus solving 6n minus 2 squared equals n minus 2 cubed gives us n equals 8.

Video Summary

The problem involves assembling \(n^3\) small cubes to create a large cube, then painting its entire outer surface. The goal is to determine the value of \(n\) when the number of small cubes with one painted face equals those with no painted faces. On each face, \( (n-2)^2 \) cubes have one face painted, totaling \(6(n-2)^2\) for the large cube. Within the cube's interior, \( (n-2)^3 \) cubes remain unpainted. Solving the equation \( 6(n-2)^2 = (n-2)^3 \) results in \( n = 8 \). Thus, the integer value of \( n \) is 8.

Keywords

cube assembly

painted cubes

unpainted cubes

equation solving

integer value n