soldiers.random - Directionless Soldiers Problem:...

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Directionless Soldiers Problem :Probability Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA squash@math.ufl.edu Webpage http://www.math.ufl.edu/ squash/ 4 May, 2010 (at 22:45 ) Soldiers. N soldiers face the sergeant, who barks “Company, right face!” Alas, the soldiers turn randomly (independently, 1 2 , 1 2 -probability) left / right. If a soldier finds himself face-to-face with another, he figures that he must have turned wrong; so he reverses (in place) to face the other way (as does the fellow he was facing —now they are back to back). Assume that the soldiers do this simultaneously, on the count of each second. A soldier who is facing no-one (he faces out, from an end of the line), or who faces someone’s back, does not turn on this count. Using notation from below, here is an example with sol- diers named 1 , 2 , . . . , 5. x : = -→ 1 - 2 - 3 -→ 4 - 5 : x 0 = - 1 -→ 2 - 3 - 4 -→ 5 x 00 = - 1 - 2 -→ 3 - 4 -→ 5 e x = - 1 - 2 - 3 -→ 4 -→ 5 The number of soldier-pairs which reverse is R ( x ) = 2
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soldiers.random - Directionless Soldiers Problem:...

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