The discussion revolves around the combinatorial problem of seating 6 men and their wives at a circular table such that no man sits beside his wife. Participants explore various approaches to calculate the number of valid seating arrangements, considering different methods of counting and subtracting invalid configurations.
Participants do not reach a consensus on the correct method or final answer. Multiple competing views and calculations are presented, with some participants questioning the validity of others' approaches.
Participants express uncertainty about the assumptions underlying their calculations, particularly regarding the treatment of couples and the implications of circular arrangements. There are unresolved mathematical steps and varying interpretations of the problem's constraints.
Individuals interested in combinatorial mathematics, particularly those exploring seating arrangements, permutations, and the complexities of circular arrangements in a mathematical context.
mattmns said:
But the men and women ARE numbered right? So permuting (or is it permutating?) the men will give a different arrangement and permut(at)ing the women will as well.Rogerio said:
Gokul43201 said:
Gokul43201 said:
abc said:
hemmul said:
Rogerio said:
whoops! so i misunderstood something...b0mb0nika said:
I don't believe you. There is much involved in subtracting the possibilities that 1, 2, 3, 4, or 5 couples could be seated together and a lot of room for error. Let's see your work.b0mb0nika said:
Rogerio said:
Number of people Solutions
4 2 (1)
6 32 (16)
8 1488 (744)
10 112512 (56256)
12 12771840 (6385920)// round_table.c
// solves the puzzle: how many ways can 6 couples sit around a round table without any man sitting next to his wife?
// ceptimus: 2005-01-22
#include "stdio.h"
// the size of the table (number of chairs, and number of people)
#define PLACES 12
int chair[PLACES]; // 0 indicates that a chair is empty, or 1 to PLACES says which person is occupying it
unsigned long solutions; // counts the number of solutions found
void seat(int person) // sit [person] somewhere (if possible) and move onto next person
{
int position;
int places;
if (person == 2)
places = 1 + PLACES / 2; // only try the first wife on one side, or opposite her husband
else
places = PLACES; // try all the other people in each available place
if ((chair[PLACES / 2] == 2) && (person == 3)) // if the first wife is seated opposite her husband
places = PLACES / 2; // only try the second husband on one side of the table
// above lines just eliminate mirror image solutions. To count such solutions, remove // from following line
// places = PLACES;
for (position = 1; position < places; position++)
{
if (!chair[position]) // if the chair is empty
{
if (person % 2) // if person is an odd number (i.e. a man)
chair[position] = person; // seat him
else // a woman
if
(
(chair[(position - 1) % PLACES] != (person - 1)) && // her husband isn't in the chair to the left
(chair[(position + 1) % PLACES] != (person - 1)) // or in the position to the right
)
chair[position] = person; // seat her
if (chair[position]) // if the person has been seated
{
if (person == PLACES) // if all the people have been seated
solutions++; // we've found and extra solution
else
seat(person + 1); // try to seat the next person
chair[position] = 0; // make the person stand up again, so we can search for further solutions
}
}
}
}
int main(int argc, char* argv[])
{
int i;
for (i = 0; i < PLACES; i++)
chair[i] = 0; // all chairs are empty
solutions = 0L; // no solutions found yet
chair[0] = 1; // sit the first man in the first chair
seat(2); // sit his wife somewhere (and then try to seat all the remaining people)
printf("%lu solutions found for seating %u people\n", solutions, PLACES);
return 0;
}