Grades 7-8 Video Solutions 2023-2023_7-8

Video Transcription

Question 24. Sophia wants to write the numbers 1 to 9 in the regions of the shape shown so that the product of the numbers in any two adjacent regions is  not more than 15. Two regions are said to be adjacent if they have a common edge. In how many ways can she do this?  Take a close look at our shape. You notice some things that stick out. First, there are two squares on the left and right side. Now since the numbers 8 and 9  cannot be multiplied by anything other than 1 without going over 15, the 8 and 9 as well as 1 will have to be here. Now it is possible to switch the placement  of 9 and 8 without it making a difference. So that is two possible options. Next, we have the next biggest numbers 6 and 7. These can only be  multiplied by 2 without going over 15. So 6 and 7 will have to be on next to each other with the 2 in between. Now the 6 and 7 could be swapped. They could also  be on the bottom or on the bottom swapped. So that is four potential options. Finally, we are left with the numbers 4, 5, and 3, which will either be  4 and 5 or 5 and 4 depending on where 6 and 7 are. So that has two more options. If we multiply all these potentials out, we'll get our answer which will be 16e.

Video Summary

Sophia wants to arrange numbers 1 to 9 in regions of a shape so that the product of numbers in adjacent regions doesn't exceed 15. Key constraints focus on fixing 8, 9, and 1 in non-overlapping positions because they can only pair with 1. This allows for swapping, giving two possibilities. Similarly, 6 and 7 must be paired with 2, which provides four arrangements (due to their swappability and position). Remaining numbers 3, 4, and 5 allow for two configurations based on previous placements. Multiplying these possibilities results in a total of 16 valid configurations.

Keywords

number arrangement

product constraint

non-overlapping positions

valid configurations

swappability