I'm totally lost in this riddle, does it exist a way to calculate the different ways to read a word using a systematically approach?. In my initial attempt what I tried to do is drawing 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 twice the word, hence I couldn't even understand if my attempt was right.
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. (Spanish word for two sides). Compute the number of different ways and joining neighboring letters can be read the phrase DOS BANDOS.
The given alternatives in my book are:
$\begin{array}{ll} 1.&1536\\ 2.&1280\\ 3.&256\\ 4.&768\\ 5.&1024\\ \end{array}$
So as mentioned, from what I could identify immediately was seen from the top to bottom there are four cubes which form vertically the word being asked. From left to right, and then from right to left another two. This accounts for six. My findings are pictured in the diagram from below, colored with orange.
But that's how far I went. As the more I looked at the stack I started to get confused on which can be allowed ways and which do already counted. Needless to say that the number I found is way off from the existing alternatives.
Hence can somebody help me with this riddle? An answer which would assist me the most is a way to methodically to solve this rather than just drawing circles over words as if it were a word search puzzle on a newspaper.
Overall does it exist a way?. Can somebody help me to be on the right path on this one?. If a diagram or drawing is necessary for an explanation or a justification of the method please include in your answer because I believe an answer for this question would be greatly improved with a visual aid, as I am not good at understanding just plain words or straightforward formulas.
The simplest way to do this to notice that:
The only way to get the correct sequence of letters is to start at the top and take a letter from each row in turn.
In making your way down row by row from the top, you always have a choice between taking the left or right letter immediately below.
You make this choice $8$ times, and have $4$ possible starting points, so the total number of routes down is
$$4\cdot 2^8=1024.$$