I am pretty sure that I have interpreted the problem wrongly and the confusingness of the paragraph is the result. (The only non-trivial interpretation which occured to me yesteday was that Omega is scanning a set of people and is changing actual answers of those who obtained "even" based on instructions given by those who obtained "odd", which was, in hindsight, quite absurd way to understand it.)

See also my last reply to Vladimir Nesov in this thread.