Closed tour of a Knight

For discussing Olympiad Level Combinatorics problems
tanmoy
Posts: 288
Joined: Fri Oct 18, 2013 11:56 pm

Closed tour of a Knight

When a Knight moves in chess, it can move to a square that is away two squares horizontally and one square vertically,or two squares vertically and one square horizontally.A closed tour for a Knight is a sequence of moves,which have the Knight start and end in the same square and visit each other square exactly once.On a $4 \times 3$ board,is there a possible closed tour for a Knight?
"Questions we can't answer are far better than answers we can't question"

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

Re: Closed tour of a Knight

Hint
$4×3$ Chess Board

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

Re: Closed tour of a Knight

Ignore the above solution.
Correct Solution

Let the tour is possible.
Total move $=12-1=11$
In every move it alters the color of the square.If the starting square is black,the ending square is not black.(Imagine it as a traditional chess board)
But,move number is odd.