There is a group of N soldiers arranged in a straight line, standing side by side. Soldier number 1 is at the extreme left and soldier "N" is at the extreme right. Each soldier has a rifle that can be fired only once, a primitive timer, understands a finite list of commands, and can exist in a finite number of states, like a finite state machine.

Each soldier has the ability to communicate only with the two adjacent soldiers, and has no means of communication with more distant soldiers. The i-th soldier can not see or hear any signals given by the (i+2)th soldier, for example. There are no radios, cell phones, or megaphones.

Your mission as the commander is to devise an algorithm by which all soldiers fire their weapons simultaneously. Soldiers 1 and N are aware of the fact that they are "different" in that they each have only one neighbor. Other than that, however, the soldiers are initially all identical. The algorithm has to work for any value of N>2.

The primitive timers are synchronized and tick off once a second. Once a soldier receives new information, the earliest he can respond in any way is on the next tick of the clock. (I would say he/she, except that they are all identical). A soldier can give a signal to each neighbor simultaneously, based on the information he received one tick earlier. Whenever a soldier's state changes, his neighbors are aware of this one tick later. At time=0, soldier 1 is given the command to start the firing procedure

1. Devise an algorithm that results in all N soldiers firing simultaneously
2. As a function of N, how many clock ticks does this take?

(In reply to re: More Information - another question by nikki)

In response to nikki's question about scenarios X vs Y:

I think of it this way.  The clock ticks, each guy is in a certain state, the Nth guy passes info to each of his neighbors and vice versa, including info about their present state.  But the info N receives doesn't include anything that the N-2 guy told the N-1 guy during this tick (or that N+2 told N+1).  No info passes about what their state change is going to be.  The soldiers individually figure out what state they should change to.  Then the last thing that happens at the end of this cycle is:  everyone changes state.  Then the clock ticks and a new cycle begins.

The Nth guy does know the rules governing state change, but since he has incomplete knowledge of the communication his neighbors are getting from their neighbors, he can't predict what state his neighbors are going to change to at the end of this cycle.

Posted by Larry on 2004-08-28 10:45:29