IBM Ponder This #143 · March 2010

Three people named Erne, Neer, and Rene are standing in a circle. Two of them hold a phone and talk to one another -- Neer is holding the first phone and Erne is holding the second phone. After they talk, Erne passes his phone to Rene and now Neer talks to Rene. After that, Neer passes his phone to Erne and the last pair, Rene and Erne, speak to one another. At this point, all three pairs have talked, each one exactly once. If we write the initial letter of the people who talked in order by phone, it looks like this:

1. The initial holder of the first phone (Neer) = N
2. The initial holder of the second phone (Erne) = E
3. The final holder of the first phone (Erne) = E
4. The final holder of the second phone (Rene) = R

We get NEER, one of the names in the list. It's easy to verify that Rene can not be a solution while Erne can be; and there is only one possible scenario of phone-passing for each of the two names.

Now consider the list of 21 people below, who are all standing in a circle, when moving clockwise their names are sorted in alphabetical order. Our riddle this month is to find out how many possible scenarios of 209 phone passes exist that spell out each name in that list, based on the initial letter of the names of the four people who speak -- the initial holders of the first and second phone, and the final holders of the first and second phone, in that order. On each phone pass, one phone is passed to one of a person's two neighbors in the alphabetically ordered cycle. In each scenario of 209 passes, every pair of people talks exactly once.

The 21 names are Adam, Boga, Dave, Eric, Fred, Gale, Ioan, John, Karl, Luke, Mark, Nick, Oded, Phil, Rani, Siva, Tony, Udit, Wolf, Yury, and Zhou.

Bonus question - what is common to all these names?

Solution

Best opened after a real attempt

To be added.