Grades 11-12 Video Solutions 2024-2024_11-12

Video Transcription

John has a number of unit cubes. Each of them is either all black or all white. He wants to build a 3x3x3 cube using 27 of them.  He wants the surface to be exactly half black and half white. What is the smallest number of black cubes he can use?  We know that the surface area of the cube will be 6 times 3 squared, which is 54. And thus, we want 27 faces white and 27 black.  We know that each of the 8 corner cubes will have 3 outward faces, and each non-corner edge cube has 2, while each center cube has 1.  If all 8 corner cubes, 1 non-corner edge cube, and 1 center face cube are black, then there will be 27 black faces. There is no way to use a smaller number of black cubes,  because we chose the sides that would maximize the number of blacks while minimizing the cubes actually used. So the minimum of black cubes necessary is 10.

Video Summary

John needs a minimum of 10 black cubes to construct a 3x3x3 cube with half of the 54 surface faces black and half white. Each corner cube, having 3 outward faces, contributes significantly to the black surface area. By using all 8 corners as black cubes, alongside 1 non-corner edge cube and 1 center face cube, John achieves the desired balance of 27 black and 27 white surface faces while minimizing the number of black units used. This configuration efficiently maximizes black surface coverage with the least number of black cubes.

Keywords

3x3x3 cube

black cubes

surface faces

corner cubes

cube construction