The Dinner and Dialogue Club has planned a series of small meetings. Each meeting would consist of two or three members enjoying friendly conversation with each other while eating food from different places all over the world. Each member is scheduled to meet exactly four times. No pair of members will meet twice, but some pairs might not meet at all.

The first thing the club did was schedule and arrange the meetings so that each member knew whom to meet and when. When it came to choosing restaurants, someone suggested that each member eat at two restaurants with eastern food, and two with western food (each restaurant is either one or the other). They liked the idea, but to their dismay, the idea was not possible without rearranging at least some of the meetings.

What possible meeting schedule might cause this to happen? How many members are there in this club, at the least?

This reminds me of that famous puzzle "The Conversing Club 3", so I made the members number 12, as in that puzzle.

[The part of the puzzle that asks what is the fewest number of members that could have the scheduling problem, is not answered in this post.]

Let the numbers 1 thru 12 be the twelve members of the Conversing Club. If they had originally scheduled their meetings as follows, there is no way for each person to have two meals of "Eastern Food" and two meals of "Western Food":

Bad Schedule
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Meeting 1:   {2, 5, 10}
Meeting 2:   {10, 6, 4}
Meeting 3:   {9, 8, 3}
Meeting 4:   {6, 11, 12}
Meeting 5:   {11, 2}
Meeting 6:   {5, 8, 11}
Meeting 7:   {5, 6}
Meeting 8:   {12, 10, 3}
Meeting 9:   {12, 9}
Meeting 10:  {3, 5, 7}
Meeting 11:  {1, 9, 10}
Meeting 12:  {11, 9, 4}
Meeting 13:  {7, 1}
Meeting 14:  {8, 1}
Meeting 15:  {3, 4, 2}
Meeting 16:  {12, 7, 4}
Meeting 17:  {7, 6, 8}
Meeting 18:  {1, 2}

The Conversing Club needs to reschedule their meetings as follows, to meet the conditions of the 2+2 meals for each person:

Good Schedule
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Meeting 1: (Eastern food)  {4, 12, 10}
Meeting 2: (Eastern food)  {11, 8}
Meeting 3: (Eastern food)  {3, 5, 7}
Meeting 4: (Western food)  {7, 8}
Meeting 5: (Western food)  {11, 10}
Meeting 6: (Eastern food)  {9, 6, 4}
Meeting 7: (Western food)  {8, 6}
Meeting 8: (Western food)  {10, 2, 9}
Meeting 9: (Eastern food)  {12, 11}
Meeting 10: (Eastern food) {7, 2}
Meeting 11: (Western food) {6, 7}
Meeting 12: (Eastern food) {5, 2}
Meeting 13: (Western food) {4, 11, 5}
Meeting 14: (Western food) {1, 12}
Meeting 15: (Western food) {3, 4}
Meeting 16: (Eastern food) {1, 10, 6}
Meeting 17: (Eastern food) {8, 3}
Meeting 18: (Western food) {2, 12}
Meeting 19: (Eastern food) {9, 1}
Meeting 20: (Western food) {3, 1}
Meeting 21: (Western food) {9, 5}

 

Edited on November 5, 2005, 4:08 pm

Posted by Penny on 2005-11-05 16:05:44