This was a difficult puzzle, even though it was seemingly simple. I did get attempts from a few people, but the only correct answer came from Gaurav S (founder of d8i.ai) – thanks and well done Gaurav!
The site from where I took this puzzle had many correct answers, most of these were largely a bit of trial and error. One solution, which I modified a little, is as follows:
For each of the output node, if the desired input for firing is X, Y, Z (each of these being either a 0 or a 1), design the weights as:
-1 if the input should be 0
1 if the input should have been 1
When you do this, you get an input output matrix as follows:
Puzzle #198 Solution
If we keep the threshold to be just below the thresholds mentioned in the table, you will get the desires output.
Hope you all enjoyed the solution!