Grades 9-10 Video Solutions 2014-Levels 9&10 Video Solutions 2014 problem22

Video Transcription

Question number 22. Ken wants to add some line segments to the picture shown so that each of the seven points here is connected to other points with exactly  the same number of connections. What is the least number of line segments that Ken must draw? So let's keep track of the connections and let's keep track of the  points. I will just label them here A, B, C, D, E, F, and G in any order really and then keep track of the number of connections each point has. So point A is  connected only to C, point B only to F, point C only to A, D is only connected to E, E is connected to both F and D, so we would put a 2 here, F is connected to  B, E, and G, so that's 3, and G is only connected to F. And so this is the initial picture. And what we want is the number of connections to be equal across  the row. So what happens, what's the equivalent of incrementing here by one?  If I do 2 here and 2 here and remain unchanged for the other values, that's the equivalent of connecting A with B. So I can draw here a segment from A to B  and that's what happens to the new number of connections. A has one more and B has one more. So I can just keep doing that. If I connect B with C, for example,  then I will have one more connection for B and one more for connection for C, while the others remain unchanged. Then continuing thus, we will increment  pairwise two at a time, that is, these numbers until we find a row with all numbers being the same. So let's see, I will just pick the two lowest numbers  here each time, so I will start here and add one, the other numbers remain the same. And then I pick the lowest pair, so maybe the first two and the next two, and  then the other numbers remain the same. And then let's go over here, increase these to threes, the other numbers remain the same, and then maybe increase this  two to a three, and then three to a four, the other numbers remain the same. And then we keep going, increasing like so, four, four, three, three, four, four, and four. And then we  see that we have here one more pair to increase, so that will happen in one, two, three, four, five, six, seven, eight, plus one for the two remaining threes, in a  total of nine steps, and steps are nine segments that have to be drawn in. So that is what Ken has to do, he needs to add nine line segments, and then each of  his points will be connected to four other points exactly.

Video Summary

Ken needs to add line segments to his diagram to ensure that each of the seven points has the same number of connections. Initially, each point varies in the number of connections it has. By strategically adding line segments, Ken incrementally increases connections in pairs of points until all points have identical connection numbers. Through a series of nine steps, each adding a new line segment between pairs of points with the fewest connections, Ken ensures that all points are ultimately connected to four others, achieving the desired uniformity. Thus, Ken needs to draw a total of nine new line segments.

Keywords

line segments

connections

diagram

uniformity

points