Finding the no. of decoders required in cascading

Question

I am trying to solve the following question.

How many 3 to 8 decoders with an enable input are needed to construct a 6 to 64 line decoder without using any other logic gate.

I have a Digital Electronics textbook in which the author gives a simple cookbook method :

If n is the number of input lines in the available decoder and N is the number of input lines in the desired decoder, then the number of individual decoders required to construct the desired decoder circuit would be \$2^{N−n}\$.

With above explanation, the answer would be 8. Then I checked the answer & it was given as 9. I have only answer & no solution & even answer also may be given wrong.

After bit of struggling, I come up with this design. Let there be 8 decoders cascaded. The 3 inputs of each of decoder would serve as inputs for target decoder. And one more 3 x 8 decoder connected to each of these 8 decoders enable line. So that makes 9 decoders.

Here is my design:

I would like to know:

1. Is my approach correct?
2. Where did I do mistake in understanding from textbook?
3. Most importantly, how to tackle problems like these. If inputs are in large number, it would be very difficult for me to come up with a design. Is there any standard formulae or any method?

Answer 1

1. Your approach is correct.

2. This is probably an error, or oversight, in the textbook: the authors forgot to allow for selecting the final decoder, which is what your 9th decoder does.

3. There's not a standard formula. First, you have to stack up enough decoders to give you the number of output pins you need. Then, you have to "tree up" enough additional decoders to select the output-layer decoders.