Six brothers Al, Bob, Cal, Don, Elmer and Fred always take the same seat when they have their meals in the circular dinner table. The following figure displays their seat numbers.
1
6 2
5 3
4
The following is known:
- Bob's seat number is precisely 1 larger than that of Elmer's seat number.
- The seat of Don is separated from Al's seat by precisely one of the other brothers.
- Al's seat number is either 1 larger than or 1 smaller than Fred's seat number.
- The absolute difference of Cal's seat number from that of Bob's and the absolute difference of Cal's seat number from that of Don's are respectively either, 2 and 5; or, 5 and 2.
Determine the respective seat numbers of the six brothers.
a) starting with clue 4, the only way to have a difference of 5 is between seat 6 and 1. So, Cal is in 1 or 6 and either Bob or Don is in the other. The only possibilities are:
B C D
1 6 4
4 6 1
6 1 3
3 1 6
b) by clue 1, Elmer's seat is one less than Bob's. This rules out the first possibility and leaves three options:
B C D E
4 6 1 3
6 1 3 5
3 1 6 2
c) By clue 3, the remaining two seats, belonging to A and F, must be adjacent. This rules out the first two options, leaving only one:
B C D E
3 1 6 2
d) And by clue 2, Al must be in 4 and Fred in 5, since Al does not sit next to Don.
So the problem is solved, and I confirm Daniel's computer solution.
I have used all the clues, but I did not need all the information in all the clues. In clue 2, I did not need the separation between Al and Don. It would have been sufficient to know that they did not sit adjacent to each other. And in clue 3, it would have been sufficient to know that Fred and Al were adjacent; the clue rules out the possibility that they were in seats 1 and 6.
In other words, the problem would be a little more difficult if clues 2 and 3 were replaced by a single clue: Al sat adjacent to Fred, but not adjacent to Don.
Edited on May 25, 2009, 3:29 pm
Analytical solution (spoiler)
