**Puzzle 181**

This is not a solution but an appendix. I drew out how boolean matrix multiplication works based on the formula :

\[ (\Psi\Phi)(x,z) = \bigvee_{y \in Y} \Phi(x,y) \wedge \Psi(y,z) \]

It's color coded (red meet and blue join) so that the you can the relationship between all three versions (1. formula, 2. feasibility relations, 3. matrices).

![boolean matrix multiplication](http://aether.co.kr/images/matrix_multiplication.svg)

You can easily see how this would work with \\(\mathbf{Cost}\\). The boolean version gives the backbone for interaction with the bottom element and the rest is just number multiplication.