Grades 9-10 Video Solutions 2014-Levels 9&10 Video Solutions 2014 problem26

Levels 9&10 Video Solutions 2014 problem26

Video Transcription

Video Summary

The problem involves counting the number of unique triangles formed by any three vertices of a cube where the vertices do not all lie on the same face. Initially, there are 8×7×6 combinations of picking any three vertices, but some are overcounted and some form triangles on a single face. The solution involves considering rotational symmetry, indicating that configurations of triangles not confined to one face repeat six times. Subtracting the 24 triangles confined to a face, the correct number of triangles meeting the conditions is 32. Thus, the solution to the problem is 32, corresponding to answer C.

Keywords

unique triangles

cube vertices

rotational symmetry

combinatorial geometry

triangle configurations