Question 44
$n$ couples attend a royal banquet; the husbands sit at one round table and the wives at another round table. The King and the Queen (not among these $n$ couples) are going to shake hands with them. Suppose the King starts shaking hands with a certain man, and the Queen starts with that man's wife. Consider two procedures:
(i) The King shakes hands with all the men in clockwise order. Whenever the King shakes hands with a man, the Queen moves clockwise to that man's wife and shakes hands with her. Suppose that when the King returns to the first man, the Queen has gone around her table $a$ times.
(ii) The Queen shakes hands with all the women in clockwise order. Whenever the Queen shakes hands with a woman, the King moves clockwise to that woman's husband and shakes hands with him. Suppose that when the Queen returns to the first woman, the King has gone around his table $b$ times.
Find the largest possible value of $|a - b|$.
(i) The King shakes hands with all the men in clockwise order. Whenever the King shakes hands with a man, the Queen moves clockwise to that man's wife and shakes hands with her. Suppose that when the King returns to the first man, the Queen has gone around her table $a$ times.
(ii) The Queen shakes hands with all the women in clockwise order. Whenever the Queen shakes hands with a woman, the King moves clockwise to that woman's husband and shakes hands with him. Suppose that when the Queen returns to the first woman, the King has gone around his table $b$ times.
Find the largest possible value of $|a - b|$.