Combinatorics on Words

1) Suppose you have to guess a 3 digit binary (i.e. 0's and 1's) code on a keypad. a) How many different codes are possible? b) Suppose that the door opens as soon as the 3 digit codes is entered. For example, if the code is 000, the door opens after 1000 is entered. Try to come up with the shortest binary sequence that is guaranteed to open the door. For example, if we had a 2 digit code, the sequence 00110 works. c)* Start exploring codes of length 4, length 5, etc. 2) Below are two directed graphs (A, B). (Note a vertex can have an edge to itself.) A Eulerian cycle is a path through ALL of the edges in a graph (using each only once) which starts and ends at the same vertex. For example, aedcb is an Eulerian cycle of graph A. a) Find all of the Eulerian cycles of graph A. Why is this not as hard as it seems? b) Find 3 different Eulerian cycles of graph B, all starting with a. Argue that there are at least 24 different Eulerian cycles of graph B. (In fact, there are exactly 24 different Eulerian cycles of graph B.)