Grades 11-12 Video Solutions 2023-2023_11-12

Video Transcription

Video Summary

The problem involves placing integers 1 to 9 in boxes such that any three consecutive numbers sum to a multiple of 3. Utilizing modular arithmetic, numbers must satisfy 0, 1, and 2 mod 3 in sequence. Numbers 7 (1 mod 3) and 9 (0 mod 3) are pre-placed, determining the necessity for 2 mod 3 to follow. With fixed mods, there are 2 factorial ways for 0 mod 3 integers, 2 factorial for 1 mod 3, and 6 ways (3 factorial) for 2 mod 3 integers. By counting principle, the total arrangements are 24 ways to fill the boxes correctly. The answer is 24.

Keywords

integer placement

modular arithmetic

consecutive sum

arrangement counting

mathematical problem