Seat the same n persons at a round table on (n-1)(n-2)/2 occasions so that no person shall ever have the same two neighbours twice.
(Dudeney, 1905.)
Solved for n even. For n odd (in general) it is still unsolved.
In the next hour I plan to
• elucidate the problem using small examples,
• outline the history and present status of the problem,
• translate the problem into the language of Graph Theory , and
• state my contribution to the solution of the problem.

Why (n-1)(n-2)/2 occasions (seating orders) ?
Knights of the Round Table How many ways can we fill the seat on the King’s right? (n-1)
Once we have filled the seat on the
King’s right, how many ways can we fill the seat on the King’s left?
(n-2)
Thus, we have (n-1)(n-2) ways to fill the two seats neighbouring the
King.
Observe that this way we have counted each possible (unordered) pair of neighbours of the King exactly twice. So there are only
(n-1)(n-2)/2 sets of neighbours for the King to choose from.
The number (n-1)(n-2)/2 is called n-1 choose 2 and it gives the number of 2-subsets of an (n-1)-set.

n = 3 (n-1)(n-2)/2 = 1 Unique solution
A C B A
C
B
B
A
A
C
B
C
A
C
B
C
B
A n = 4 (n-1)(n-2)/2 = 3 Unique solution
A A A
D B C B D C
C D B n = 5 (n-1)(n-2)/2 = 6 Unique solution
A A A A
E B D B C B E C D
A
C
D C C E E D B D E B
E
C
A
B
D

Why Dudeney’s ?
Henry Ernest Dudeney (1857-1930) "England's greatest composer of puzzles.” Some of his puzzles inspired serious mathematical research. Had only basic education, self-taught in math and chess. He was publishing puzzles in a local (Mayfield, Sussex) paper at the age of 9 and later in London journals for the rest of his life.
S E N D
M O R E
M O N E Y
Books : The Canterbury Puzzles (1907), Amusements in Mathematics (1917),
Modern Puzzles (1926).
The history of mathematical puzzles entails nothing short of the actual story of the beginnings and development of exact thinking in man
.
He first published the round table problem in 1905. In 1917 he published it in his puzzle book, Amusements in mathematics, as Problem 273. Hence his name became associated with the round table problem .

Judson’s round table problem ?
Seven persons met at a summer resort, and agreed to stay as many days as there are ways of sitting at a round table, so that no one shall sit twice between the same two companions. They remained 15 days. It is required to show in what way they may have been seated.
( Judson, 1899 , Am.Math.Monthly)
Charles H. Judson, the first professor hired by Furman University , 1850 . He taught mathematics and science.
He became president of the Female
College of Furman Univ.
in 1866.
He appointed his sister as " lady principal .” She “ sparked generations of young women with the desire to excel and a belief in their own potential.
”
The 1887 Furman faculty

Chronology of partial solutions to the round table problem
1900 Judson: n = 6,8
1900 Dickson: shows two different solution at n = 6
1904 Safford: n = 7
1905 Dickson: n = 10,12
1917 Dudeney: n = 11
1964 Kotzig n = p+1 (p stands for prime number)
1980 Nakamura, Kiyasu, Ikeno: n = p k +1
1980 Nakamura, Kiyasu, Ikeno: n = 19, 23
1991 Heinrich, Kobayashi, Nakamura: some cases of n = p+2
1993 Kobayashi, Kiyasu, Nakamura : even number of people
2002 Kobayashi, Akiyama, Nakamura: further cases of n = p+2
2002 Varga (helped by a computer) : n = 27, 29, 35, 37
At present: n = 41 is the smallest case without a known solution .

Some terminology from Graph Theory
A graph is a set of points and edges . (An edge is a line-segment connecting two distinct points of the graph.) A graph is complete , if any two distinct points are connected by an edge. A complete graph with n points is denoted by Kn. For example, here is a K
4
:
D
A
C
A 2-path in a graph is a path of length 2. (Two distinct edges with a common point.)
B
A cycle of length n (formed by n distinct points and n distinct edges) in a graph with n points is called a Hamilton cycle . The red edges in the K
4 on the left form a Hamilton cycle.

Translation of the round table problem to Graph Theory
Find a set of (n-1)(n-2)/2 Hamilton cycles in K n property that every 2-path of K n with the appears in exactly one of the cycles in the set .
A A A
D B C B D C
C
A
D
A
B
A
D
C
B D
C
B D
C
B

My bit of contribution: Impose an extra requirement on the round table problem
Find a set of (n-1)(n-2)/2 Hamilton cycles in K n property that every 2-path of K n with the appears in exactly one of the cycles in the set .
Further, rotating any one of Hamilton cycles in the set must yield an other Hamilton cycle in the set.
The extra requirement reduces the size of the search space from order n^(n 3 ) to order n^(n 2 ) and only (n-1)/2 cycles need be found (for odd n).
Conjecture: Solutions to the round table problem satisfying the extra requirement exist for every n, n=3,4,5,…, except n=7.