[superprismatic]

A benevolent king hosts wedding

anniversary dinners for some of

his subjects on several days

each year. He has a 61-seat

round table at which to seat

30 couples who are celebrating

their wedding anniversaries on

the day of the dinner. He has

decreed that each couple will

be seated N-seats away from each

other if they are celebrating

their N^th anniversary.

So, for example, if a couple

has been married 1 year, they

will be seated next to each

other; married 2 years and they

will have one person between them;

etc. The king will sit at the

one remaining seat. Imagine that

you are the king's chief of

protocol and you are given the

task of arranging the seating

for the 30 couples with the

following anniversary distribution:

# couples   years married

    2           2

    2           4

    1           5

    1           6

    2          10

    2          11

    3          12

    1          14

    1          18

    1          19

    2          21

    3          23

    1          24

    1          27

    4          28

    3          29

[/code]

Being a Mathematics buff, you

realize that this can be done

because 61 is a prime.  So, in what

order do you seat the 60 guests?

[bonanova]

Clarify:

.

1. Since the table is round, between any two persons there are two sets of intervening people.
   Do we always take the smaller number?
   Doh! Forget that one ... but,
   .
   
2. The King sits after all the spacings have been set.
   So the King does not apply to the separation count?

[superprismatic]

Clarify:

.

1. Since the table is round, between any two persons there are two sets of intervening people.
   Do we always take the smaller number?
   Doh! Forget that one ... but,
   .
   
2. The King sits after all the spacings have been set.
   So the King does not apply to the separation count?

To be specific, if a couple are on either side of the king, they are celebrating their 2^nd anniversary.

So, the king counts in the separation. I hope this clears it up.

[Guest]

So, in what

order do you seat the 60 guests?

You seat them in the order that they arrive.

[superprismatic]

You seat them in the order that they arrive.

I didn't mean in time order. How do you order them around the table?

[Guest]

If he is hosting several different parties do they need to be seated apart according to their anniversary at all parties or just at the party for their anniversary?

Edited August 21, 2009 by Chuck2177

[superprismatic]

If he is hosting several different parties do they need to be seated apart according to their anniversary at all parties or just at the party for their anniversary party?

This is just for one of his many parties. All attendees are celebrating their anniversaries on the date of the dinner. The poor chief of protocol has to make the seating chart for all the other dinners as well! If you can find a good algorithm for him to use, you will simplify his life enormously!

[bonanova]

A previous relates to this one, but with a difference.

Mine is linear, has one pair of each type, and fewer pairs.

I've been trying to use it as a first step, without success.

Still, it might provide a clue.

3. 3 1 2 1 3 2
4. 2 3 4 2 1 3 1 4
6. 6 1 5 1 7 3 4 6 5 3 2 4 7 2
8. 8 6 4 2 5 7 2 4 6 8 5 3 1 7 1 3
11. 1 2 1 11 2 3 9 10 4 3 8 5 7 4 6 11 9 5 10 8 7 6
12. 1 2 1 3 2 12 10 3 11 4 5 9 6 8 4 7 5 10 12 6 11 9 8 7
15. 1 2 1 3 2 4 14 3 15 13 4 5 12 6 7 10 11 5 8 9 6 14 7 13 15 12 10 8 11 9
16. 1 2 1 3 2 4 16 3 13 5 4 15 12 14 6 5 7 8 11 9 10 6 13 16 7 12 8 15 14 9 11 10

[Guest]

[5, 12, 23, 10, 28, 4, 11, 11, 12, 4, 28, 19, 21, 12, 12, 27, 29, 11,

11, 21, 12, 18, 29, 24, 23, 23, 12, 28, 10, 28, 29, 0, 2, 21, 2, 23,

14, 28, 10, 18, 21, 4, 2, 28, 2, 4, 6, 23, 29, 27, 14, 29, 6, 19, 10, 28, 5, 28, 23, 29, 24]

basically i wrote an algorithm that tries random positions until it finds one that works.

(the 0 would be the king)

[superprismatic]

[5, 12, 23, 10, 28, 4, 11, 11, 12, 4, 28, 19, 21, 12, 12, 27, 29, 11,

11, 21, 12, 18, 29, 24, 23, 23, 12, 28, 10, 28, 29, 0, 2, 21, 2, 23,

14, 28, 10, 18, 21, 4, 2, 28, 2, 4, 6, 23, 29, 27, 14, 29, 6, 19, 10, 28, 5, 28, 23, 29, 24]

basically i wrote an algorithm that tries random positions until it finds one that works.

(the 0 would be the king)

I did a depth-first tree search. When I reorder the couples to place, I find that I can get very many solutions. I had hoped that someone would come up with a more enlightened approach than the ones you and I found.