# What makes a path finding heuristic Monotonic vs. Non-monotonic?

I am currently working on an algorithm that solves a path through the maze. So far I have implemented a Manhattan and Pythagorean Heuristic. I was wondering what makes a heuristic monotonic and how would one go about making a heuristic non-monotonic.

A heuristic $$\h\$$ is monotonic when, in addition to being admissible (meaning the estimate is always equal or lesser to the actual minimum cost) the heuristic satisfies the relation:

$$h(x) <= d(x,y) + h(y)$$

fo all adjacent nodes $$\x\$$ and $$\y\$$, where $$\d(x,y)\$$ is the distance from node $$\x\$$ to node $$\y\$$.

More can be found here.

Update: (thanks to ilmari-karonen):