jmc

We can simply see that path $A\to X\to M\to Y\to C\to Z$ will give us the smallest value. Adding, $5+2+6+5+10=\boxed{28}$. This is nice as it's also the smallest value, solidifying our answer. You can also simply brute-force it or sort of think ahead - for example, getting from A to M can be done $2$ ways; $A\to X\to M$ ($5+2$) or $A\to M(8)$, so you should take the shorter route ($5+2$). Another example is M to C, two ways - one is $6+5$ and the other is $14$. Take the shorter route. After this, you need to consider a few more times - consider if $5+10$ ($Y\to C\to Z$) is greater than $17(Y\to Z$), which it is not, and consider if $25(M\to Z$) is greater than $14+10$ ($M\to C\to Z$) or $6+5+10$ ($M\to Y\to C\to Z$) which it is not. TLDR: $5+2+6+5+10=\boxed{28}$. [Note: This is probably just the thinking behind the solution.] {Double-note: As MaxyMoosy said, since this answer is the smallest one, it has to be the right answer.}