Answer to Question #464 in Statistics and Probability for olaf

Question

Three brothers visited their sick friend in one day, and in the same day their wives also visited him. No one of the visitors came more than once. Every brother has met both of his brothers’ wives in the house of their sick friend. Prove that at least one of the brothers have also met his wife it that house.

Answer 1

Suppose that brother A did not met his wife in the house of his sick friend. Then his wife either has left before he came (in this case she has also left before the wives of two other brothers left, because, according to the task, her husband met them) or has come after he had left (in this case she has also come after the wives of two other brothers). Thus every wife could not meet her husband only if she either has left before the wives of other brothers or has come after them. However, out of three women at least one has not come after two others and has not left before them. Therefore, at least one of three wives has met her husband at their friend’s house.