Question

Six robots $A,B,C,D,E,F$ can contact as follows (directed): $B\!\to\!E$, $C\!\to\!B$; $A\!\to\!D$, $E\!\to\!D$, $F\!\to\!D$; $F\!\to\!C$; $D\!\to\!C$, $D\!\to\!E$; $A\!\to\!F$. Robots can relay messages along allowed contacts. What is the {minimum number of robots between $A$ and $B$ to send a message from $A$ to $B$?}

Hint:

In directed-contact puzzles, lock the mandatory final hop first, then work backwards to the source to find the shortest chain.

Answer 1

Step 1: Observe the required last hop.
$B$ can be contacted by $C$ (given). So any path to $B$ must arrive via $C$.
Step 2: Reach $C$ from $A$.
From $A$ we have $A\!\to\!F$ and $A\!\to\!D$. Either of these can reach $C$: $F\!\to\!C$ and $D\!\to\!C$.
Step 3: Construct a shortest path.
A valid shortest route is \[ A \;⇒\; F \;⇒\; C \;⇒\; B, \] which uses exactly two intermediate robots $(F,C)$. (Path $A\!⇒\!D\!⇒\!C\!⇒\!B$ is another with two.) No single robot from $A$ contacts $B$ directly, and $B$ must be entered via $C$, so fewer than two intermediates is impossible.
[2pt] \[ \boxed{2} \]