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Perhaps they all need to be counters? How do they know k? Couldn't the king simply set the chalice to whatever state it was in before calling in each prisoner? That way, it would always look like it was untouched by anyone else to the current prisoner. Or is that not true if they know k? My brain hurts.

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K is known by the prisoners. The king can change it after he calls in the prisoner, up to a maximum of k times. The king must call every prisoner any arbitrary number of times, so we can always set that number of times to be greater than K.