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

Video Transcription

Christina has a set of cards numbered 1 to 12. She places 8 of them at the vertices of an octagon so that the sum of the numbers at the endpoints of each edge is a multiple of 3.  Which numbers did Christina not place? Note that if one of the numbers at a vertex is a multiple of 3, then the number at every neighboring vertex must also be a multiple of 3.  Because we only have 4 multiples of 3 from 1 to 12, we come to a contradiction and we can't have any multiples of 3 at any vertex. Therefore, our answer is E.

Video Summary

Christina needs to arrange cards numbered 1 to 12 on an octagon's vertices so each edge's sum is a multiple of 3. The available numbers 1 through 12 include four multiples of three: 3, 6, 9, and 12. However, placing any multiple of 3 requires all neighboring vertices to also have multiples of 3, which is impossible with only four such numbers. Therefore, no multiples of 3 can be used, meaning the numbers Christina did not place are 3, 6, 9, and 12. Hence, the answer is that these four numbers were not placed.