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

Video Transcription

Jill has some black, grey, and white unit cubes. She uses 27 of them to build a 3x3x3 cube. She wants the surface to be exactly 1 third black, 1 third grey, and 1 third white.  The smallest number of black cubes she can use is A, and the largest possible number of black cubes she can use is B. What is the value of B minus A?  We know that the surface area of a cube is 6 times 3 squared, which is 54, and thus each color should have an area of 18.  In order to maximize the area, we can use edge cubes, or corner cubes. Each corner cube will have an area of 3, and each edge cube will have an area of 2.  Also, each center cube will have an area of 1, which is the smallest possible.  We know that there are 8 corner cubes, 12 edge cubes, 6 center faces, and 1 cube completely hidden from the surface.  Therefore, the smallest number of unit cubes to make 18 is to use 6 corner cubes. The largest number of cubes to make 18 is 12. We can use 6 center faces and 6 edge cubes.  We can also choose the hidden unit cube to have that same color. So we can use 13 black unit cubes, 1 on the center, 6 center faces, and 6 edge cubes.  Therefore, we have B equals 13, A equals 6, and therefore B minus A is 7.

Video Summary

To build a 3x3x3 cube with a surface exactly one-third black, grey, and white, Jill uses a total of 27 unit cubes. Each color should cover a surface area of 18 since the total surface area is 54. To minimize black cubes, she can use 6 corner cubes, achieving a surface coverage of 18. To maximize them, she uses 6 center faces, 6 edge cubes, and the hidden center cube, totaling 13 black cubes. Thus, the smallest number is 6, and the largest is 13. Consequently, the difference \( B - A \) is \( 13 - 6 = 7 \).

Keywords

3x3x3 cube

unit cubes

surface area

color distribution

difference calculation