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.
It is also assumed that every rule is to be understood as implying the words “and vice versâ.” Thus the first rule would imply the addition “and, if either Cole or Lang takes only one condiment, then Barry takes salt.”]