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ASU 238 All Soviet Union MO 1977 b/w checkers around a circle

Source:

July 6, 2019
circleColoringcombinatoricscombinatorial geometry

Problem Statement

Several black and white checkers (tokens?) are standing along the circumference. Two men remove checkers in turn. The first removes all the black ones that had at least one white neighbour, and the second -- all the white ones that had at least one black neighbour. They stop when all the checkers are of the same colour.
a) Let there be 4040 checkers initially. Is it possible that after two moves of each man there will remain only one (checker)?
b) Let there be 10001000 checkers initially. What is the minimal possible number of moves to reach the position when there will remain only one (checker)?