Vjylku

I have been going in circles for a few days now with this word-search puzzle and I still cannot find a clever solution to the problem. Is there a systematic approach that doesn't involve guessing? The problem is as follows:

At a kindergarten's playroom in Taichung a teacher assembled the following configuration using alphabet cubes forming a stack (see the figure as a reference) where it can be read the word DOS BANDOS (the Spanish word for two sides). Calculate the number of different ways joining neighboring letters can be read the phrase DOS BANDOS.

The possible solutions in my book are:

$begin{array}{ll}
1.&1536\
2.&1280\
3.&256\
4.&768\
5.&1024\
end{array}$

In my initial attempt what I tried to do is to draw a circle over each time I could identify the word being asked but in the end, I got very confused and I felt that I counted a possibility twice, hence I couldn't even understand if my attempt was right. During this process I could identify immediately that the word could be read from the top to bottom, there I counted four cubes, going from left to right, hence $4$ ways and from right to the left $2$ ways which together would account for $6$. This is summarized in the drawing from below where the circles are painted with blue color.

In the end that how far I went. As the more I looked at the stack I started to get confused on which zig zag lines are allowed and which do are already counted.

Therefore can somebody help me with this riddle?. To be honest I have little experience with these kinds of problems so I'd like somebody could be as much as detailed possible and include some drawing (perhaps using mine as a reference) and justify a method for solution.

It is very important for me to get a visual aid, because I really don't feel that solely a paragraph alone would be enough to understand, even as hard as I could. So really please if you can help me with this, include some sort of drawing or schematic so I can understand how to calculate the number of ways.

Again, I am looking for an answer which can solve this problem and that it can be extended to similar problems. Does it exist a way? I've been told a hint which mentions that I should consider turns to the left or right. But I don't know how to use this information.

There is also, in the bottom of the truncated pyramid it can be read the word DOS. Would this count if going from bottom to the top?.

Overall I hope somebody can take time and answer these questions. Because I really need help with this one and I'm confused.

The answer is - the number of different ways joining neighboring letters to get the phrase DOS BANDOS is 1024.


You can record your path down as a sequence of 1s and 0s - left turn 0, right turn 1. Your record from any of the 4 D's at the top could be any combination of 8 ones/zeros. It gives you a total of 256 paths. Multiply it by 4 possible Ds to start with and you get the answer. Below is an example of a path record:

                D

               /                            left 0 

              O   O 

                                          right 1

            S   S   S 

                                          right 1 

          B   B   B   B

                 /                          left 0 

        A   A   A   A   A 

                                          right 1 

      N   N   N   N   N   N 

                                          right 1 

    D   D   D   D   D   D   D

                   /                        left 0

  O   O   O   O   O   O   O   O

                 /                          left 0 

S   S   S   S   S   S   S   S   S

The solution to this problem is...

$1024$


From any given circle that is not in the bottom row, you can either go down and to the left or down and to the right. Because there are four starting points and eight rows in which we make a turn, the solution is $4 times 2^{8} = 1024$.