Grades 9-10 Video Solutions 2023-2023_9-10

Video Transcription

Video Summary

Elizabeth can arrange the integers 1 to 9 in 9 boxes such that any 3 adjacent boxes sum to a multiple of 3 in \(6^4\) ways. To achieve this, she organizes the numbers into sequences of groups like \(a, b, c, a, b, c, a, b, c\), where \(a, b, c\) represent numbers equivalent to 0, 1, and 2 mod 3 respectively. There are 6 permutations of arranging \(a, b, c\), and for each residue class (numbers 3, 6, 9 for \(a\); 1, 4, 7 for \(b\); 2, 5, 8 for \(c\)), there are 6 possible arrangements. Therefore, the total arrangements are \(6 \times 6 \times 6 \times 6 = 1296\) ways.

Keywords

integer arrangement

adjacent boxes

multiple of 3

residue classes

permutations