Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem14

Levels 9&10 Video Solutions 2010 problem14

Video Transcription

Question number 14, how many different sums can we get by adding the numbers and the direction of the arrows while moving from point A to point B?  Here in our diagram point A indicates start, we're trying to get to point B along the possible paths here indicated by the arrows.  So let me make a copy here of that diagram, let me do two copies. And what we see is that moving from the value 1 to 3 for example we have only two paths  possible but along either one of those two paths there is a 1 and a 2 that we encounter no matter how we start before we get to 3.  So we can move in the following fashion from 1 to 3 like that diagonally, it doesn't really matter which path we choose, we're going to see a 1, a 2 and a 3 that together add  to 6 and then we can keep going here diagonally again and it again doesn't matter if we go up and right or right and up from 6 we will increase this sum by 2 and by a 1 so we end  up looking at 9 over here. So that's one possible value that we encounter moving from A to B but what we have neglected  to look at are the numbers over here in green, we haven't touched those values yet. So moving from 1 to 1 across like this over here we end up making 1 plus 2 plus 1 or  a 4 and likewise over here. And then from these corners there is only one path that takes us to B so we start at  the value 4 over here, let me shade that in green and then our goal is to get to B or to that 1 over there and that is accomplished by moving here up and we have a 4 plus 2 plus  1 or moving across from here and we have a 4 plus 2 plus 1, both of these give us 7  and those are the only possible paths as should be evident here by looking at the two possibilities  so we have exactly two choices, we have a 7 or a 9 for the sums and that is answer B 2.

Video Summary

The problem involves finding the number of different sums obtained by moving from point A to point B along various paths on a diagram. By examining the paths, one possible path accumulates sums of 1, 2, and 3, leading to a total of 6, then 9 when further numbers are added. Another path sum involves 1, 2, and 1, leading to 4, and then continues to sum to 7 when reaching B. After evaluating all possible paths and combinations, the only sums achievable are 7 and 9, resulting in exactly two different sums. Therefore, the answer is B 2.