Implementing a function with 3x8 decoder and a minimal number of logical gates

Question

so I've got this question that I'm stuck on for a very long time and help would be really appreciated! It goes like this:

Implement the function: f(a,b,c,d)=∑(0,1,3,4,8,9,15) with a 3x8 decoder and a minimal number of logical gates.

I've drawn a truth table and a Karnaugh map and got to:

f= ABCD + A'C'D' + A'B'D + B'C'

(for self check I even used: http://www.32x8.com/circuits4---A-B-C-D----m-0-1-3-4-8-9-15----------------option-0)

I tried over 10 different methods and yet I cannot get rid of any variable, which I need to because I need to use the 3x8 decoder..

Thanks in advance if anyone could help with it..

Answer 1

Here's how you can do it with three additional gates (plus an inverter). You pick one of the four variables, and expand your expression w.r.t. to it. For example, if we pick $B$,

$$ (B' + ACD + A'C'D') (B + A'D + C'). $$

You feed the other three variables to the decoder. and replace each term in the expression above in $A$, $C$, and $D$ with the corresponding output(s) of the decoder.