Implementing a boolean function using at most 10 gates

Question

I was given the following question:

Given the following truth table, implement the 2 functions needed for \$cout\$ and sum (\$s\$) using only XNOR, NAND and OR (only the 2 inputs version for each gate) such that the total number of gates used is at most 10 (in both function combined). Here's the truth table:

What I tried so far was first to make a K-map for each one of the functions to get simplified expressions in terms of OR and AND gates, I got the following expressions (for means of simplification, I wrote \$cin\$ as \$c\$, and \$a\_ns\$ as \$s\$ and the output for the second function (\$s\$) is just called \$sum\$): $$ cout = bc+abs+acs+a'bs'+a'cs' $$ $$ sum = a'bc'+a'b'c+abc+ab'c' $$

also tried to write \$cout\$ as : $$ as(b+c)+a's'(b+c)+bc=(b+c)(XNOR(a,s))+bc. $$

I know how to express it with only the given gates but with way more than 10.

Any help would be much appreciated!

Answer 1

Here's a clue to get you started: -

That uses two XNOR gates to make s.