(1) If Acres is with (i.e. is in the same party with) his wife, and Barry with his, and Eden with Mrs. Hall, Cole must be with Mrs. Dix; (2) If Acres is with his wife, and Hall with his, and Barry with Mrs. Cole, Dix must _not_ be with Mrs. Eden; (3) If Cole and Dix and their wives are all in the same party, and Acres _not_ with Mrs. Barry, Eden must _not_ be with Mrs. Hall; (4) If Acres is with his wife, and Dix with his, and Barry _not_ with Mrs. Cole, Eden must be with Mrs. Hall; (5) If Eden is with his wife, and Hall with his, and Cole with Mrs. Dix, Acres must _not_ be with Mrs. Barry; (6) If Barry and Cole and their wives are all in the same party, and Eden _not_ with Mrs. Hall, Dix must be with Mrs. Eden. The Problem is to prove that there must be, every day, at least _one_ married couple who are not in the same party. pg192 6. After the six friends, named in Problem 5, had returned from their tour, three of them, Barry, Cole, and Dix, agreed, with two other friends of theirs, Lang and Mill, that the five should meet, every day, at a certain _table d'hôte_. Remembering how much amusement they had derived from their code of rules for walking-parties, they devised the following rules to be observed whenever beef appeared on the table:-- (1) If Barry takes salt, then either Cole or Lang takes _one_ only of the two condiments, salt and mustard: if he takes mustard, then either Dix takes neither condiment, or Mill takes both. (2) If Cole takes salt, then either Barry takes only _one_ condiment, or Mill takes neither: if he takes mustard, then either Dix or Lang takes both. (3) If Dix takes salt, then either Barry takes neither condiment or Cole take both: if he takes mustard, then either Lang or Mill takes neither. (4) If Lang takes salt, then Barry or Dix takes only _one_ condiment: if he takes mustard, then either Cole or Mill takes neither. (5) If Mill takes salt, then either Barry or Lang takes both condiments: if he takes mustard, then either Cole or Dix takes only _one_. The Problem is to discover whether these rules are _compatible_; and, if so, what arrangements are possible. [N.B. In this Problem, it is assumed that the phrase "if Barry takes salt" allows of _two_ possible cases, viz. (1) "he takes salt _only_"; (2) "he takes _both_ condiments". And so with all similar phrases. It is also assumed that the phrase "either Cole or Lang takes _one_ only of the two condiments" allows _three_ possible cases, viz. (1) "Cole takes _one_ only, Lang takes both or neither"; (2) "Cole takes both or neither, Lang takes _one_ only"; (3) "Cole takes _one_ only, Lang takes _one_ only". And so with all similar phrases.