# Kings, Dukes, and Truelists

Posted on: September 15, 2010

On Tanya Khovanova’s Math Blog I encountered a couple of interesting probability puzzles involving a truel. In one, the three men have an infinite number of bullets and they shoot in order until only one man is left alive, and in the other, each truelist has only one bullet. Ms. Khovanova gives the solution for the second one, based on probabilities and basic survival instincts, but leaves the first one to the reader. These puzzles remind me of a problem we were given in Professor Edgar’s M317 Proofs class:

“The five Dukes of Earl are scheduled to arrive at the royal palace on each of the first five days of May. Duke One is scheduled to arrive on the first day of May, Duke Two on the second, etc. Each Duke, upon arrival, can either kill the king or support the king. If he kills the king, he takes the king’s place, becomes the new king, and awaits the next Duke’s arrival. If he supports the king, all subsequent Dukes cancel their visits. A Duke’s first priority is to remain alive, and his second priority is to become king. Who is king on May 6?”

We all tried to solve this, and there was some interesting logic involved in our attempts, but even with the hint that working backwards would help, only one young lady in our class was able to solve it. She also solved it for six Dukes, and discovered a pattern that followed for even and odd numbers of Dukes. I have given a few hints, but I am not going to give the solution in case others might like to try it.

I have noticed that in order to solve any of these three puzzles, not only must probabilities for each action be considered, but one must have a good understanding of human nature as well. These are not merely about numbers, but also involve a fair amount of psychology. Two things I take away from this: 1) sometimes it is necessary to work backwards to find a solution, and 2) math does not exist in a vacuum!

### 2 Responses to "Kings, Dukes, and Truelists"

I love these types of questions. I can now try to solve these over the weekend. When trying to solve these problems, I think about the movie, “Patch Adams.” I think of when Robin Williams was told to “look past the problems and not at the problem.” Robin Williams was told to not look at the two fingers being held up, but instead to look at the four fingers by looking beyond the two fingers. So many times people concentrate so much at the problem that they over think it and don’t see the obvious solution.

That blog is one of my favorites. She writes very well and about interesting topics.

I am glad you noticed the connection between these problems and the proofs problem we discussed at the beginning of last semester.

If you are looking for an interesting problem, and I don’t know where it leads but there may be some interesting mathematics behind this: try to generalize the truel problem to $n$ people (start small by going to 4 people). You may have to choose the probabilities on your own, but I think it would be neat to look at.