3. A safe is locked by a combination of of four binary digits (that is, 0 or 1), but the owner has forgotten the combination. The safe is designed in such a way that no matter how many digits have...

Extracted text: 3. A safe is locked by a combination of of four binary digits (that is, 0 or 1), but the owner has forgotten the combination. The safe is designed in such a way that no matter how many digits have been pressed, if the correct combination of three digits is pressed at any point, then the safe automatically opens (there is no "enter" key). Our goal is to find the minimum number of digits that one needs to key in in order to guarantee that the safe opens. In other words, we wish to find the smallest possible length of a binary sequence containing every four-digit sequence in it. (a) Create a digraph whose vertex set consists of three-digit binary sequences. From each vertex labelled ryz, there is one outgoing edge (labelled 0) leading to vertex yz0, and another outgoing edge (labelled 1) leading to vertex yz1. (b) Explain why every edge represents a four digit sequence and why an Eulerian tour of this graph represents the desired sequence of keystrokes. (c) Find the minimum number of digits that one needs to key in to guarantee that the safe opens.