number of shortest paths possible from black triangle to bottom row

N

nanase

Junior Member
Joined
Aug 8, 2019
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232
triangles paths.jpg
Hi I need help in answering this, the answer key says B but workings and answer did not result to it.
I tried listing possible paths from current row to the next row and adding everything.
ans4.jpg
can you hint me what did I miss?
 
After 3 attempts I did get the correct answer. The question I should've asked myself: what's the maximum number of "above" neighbors a triangle can have?
 
I tried by assigning numbers, the possible shortest paths should be 3x3x3x3 =81
ans7.jpeg
what did I miss here?
 
Dr.Peterson said:
There are 3 triangles in the second row; they can be reached, respectively, in 1, 1, 1 ways.

Now use those to fill in the 5 triangles in the third row.
Click to expand...
I am not getting the logic here, If I try to understand from your input, that means in the third row ( 5 triangles ) it will all be 2, 2, 2, 2 ,2 ? since you only need 2 ways from the top black triangle?
 
nanaseailie said:
I am not getting the logic here, If I try to understand from your input, that means in the third row ( 5 triangles ) it will all be 2, 2, 2, 2 ,2 ? since you only need 2 ways from the top black triangle?
Click to expand...
I'm following the method suggested by @blamocur in #2: label each triangle with the total number of ways to get there (by a shortest path, which means one row at each step) from the top.

Here is the start of my work, going one row further, and into the next:

1699102336523.png

Each triangle has the sum of the numbers in each of the triangles it touches in the row above. Each 3 comes from 1+1+1, and the 7 comes from 1+3+3. Do you see why?

Do you see?
 
Dr.Peterson said:
I'm following the method suggested by @blamocur in #2: label each triangle with the total number of ways to get there (by a shortest path, which means one row at each step) from the top.

Here is the start of my work, going one row further, and into the next:

View attachment 36695

Each triangle has the sum of the numbers in each of the triangles it touches in the row above. Each 3 comes from 1+1+1, and the 7 comes from 1+3+3. Do you see why?

Do you see?
Click to expand...
Hi Dr. Peterson, there must be something I miss here, please enlighten me further. The first row is all 1 because it takes 1 step from the black triangle to reach the first row triangles, but why the 2nd row has 1 and 3? where is the 1 and 3 coming from?
 
in the first row (black triangle) I put 3 as there are 3 possible ways to get down
in the 2nd row I put 4 , 3 , 4 as these are the numbers of possible ways to get to 3rd row.
in the 3rd row I put 4 3 5 3 4
in the 4th row I put 4 3 5 3 5 3 4
in the 5th I am not assigning numbers because I have reached my destination.
blamocur said:
What would you put in those triangles?
Click to expand...
 
nanaseailie said:
Hi Dr. Peterson, there must be something I miss here, please enlighten me further. The first row is all 1 because it takes 1 step from the black triangle to reach the first row triangles, but why the 2nd row has 1 and 3? where is the 1 and 3 coming from?
Click to expand...
Here, in green, is the one way to get to the triangle marked 1, and, in blue, the three ways to get to the one marked 3.

1699231972122.png

The 1 is the "sum" of the value (1) in the one triangle you can get there from.

The 3 is the sum of the values (1) in each of the three triangles you can get there from.

nanaseailie said:
in the first row (black triangle) I put 3 as there are 3 possible ways to get down No, you need a 1 there (if anything) because you start there.
in the 2nd row I put 4 , 3 , 4 as these are the numbers of possible ways to get to 3rd row. No, the numbers are the ways to get there, not to go from there.
in the 3rd row I put 4 3 5 3 4
in the 4th row I put 4 3 5 3 5 3 4
in the 5th I am not assigning numbers because I have reached my destination.
Click to expand...
Apparently you misunderstood what this meant:
blamocur said:
I would mark every triangle with the number of shortest ways to get to it.
Click to expand...
 
nanaseailie said:
in the first row (black triangle) I put 3 as there are 3 possible ways to get down
in the 2nd row I put 4 , 3 , 4 as these are the numbers of possible ways to get to 3rd row.
in the 3rd row I put 4 3 5 3 4
in the 4th row I put 4 3 5 3 5 3 4
in the 5th I am not assigning numbers because I have reached my destination.
Click to expand...
I don't know how to get your approach to work, but my suggestion in post #2 was different: "...mark every triangle with the number of shortest ways to get to it.". The numbers I use are the numbers of shortest ways from the top to specific triangles. Post #8 by @Dr.Peterson shows correct numbers in the first several triangles. Do they make sense to you?
 
wow wow Dr Peterson and Glamocour after doing some re-thinking from your suggestions, I got these numbers :
ans7.1.jpeg
and adding up all the numbers in the last row, I got 153 is this correct?
I really have problems in interpreting and assigning the paths before. Thank you so much for re-explaining it.
 
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