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

Video Transcription

Problem number 28. The numbers from 1 to 11 are to be placed in the hexagons so that the sum of  the three numbers around each of the six black dots is the same. Three of the numbers have already been placed. What numbers will be placed in the hexagon with a question mark?  Let's first give each of the places in the diagram a name so that we can refer to it easily.  For the two leftmost black dots we have 6 plus a plus b equals 4 plus b plus x and that gives us  x equals a plus 2. Okay but we still have one two variables and one equation so we don't have  anything yet so we keep on going. For the two middle dots we have that a plus c plus y equals  x plus c plus d so y is equal to d plus 2. Finally for the rightmost dots we have y plus e plus question mark is equal to e plus d plus 11 so question mark equals 9.  So then we can take each of the values and solve them to plug in our values for each of the numbers.  In fact we only need the value of the question mark  and so that tells us that the question mark is equal to 9 and so the correct answer is e.

Video Summary

In the problem, numbers 1 to 11 are placed in hexagons around dots so that the sum around each black dot is the same. Starting with given placements, relationships between variables lead to solving equations. Specifically, the relationship and equating sums help determine the unknown number. Step-by-step analysis reveals that the number replacing the question mark in the hexagon is 9.