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Posts Tagged ‘**brain teaser**’

### Kings, Dukes, and Truelists

Posted September 15, 2010

on:- In: Blogs | Problems
**2**Comments

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!