Brain Teasers
Knight's Tour II
The knight left Breadland, and went to a neighboring kingdom named Cakeland. Cakeland is much similar to Breadland, but unlike Breadland, Cakeland is 12x4. Cakelanders like cake, as well as Breadlanders like bread.
The knight was on a square of Cakeland, he wanted to visit each square of Cakeland exactly once, and return to the square he was originally on. Each step, the knight moved to one of the squares nearest to that on which it stands but the queen can't reach.
The knight thought that since Cakeland has an even number of squares, he could achieve his goal this time.
However, the knight still failed.
Do you know why?
The knight was on a square of Cakeland, he wanted to visit each square of Cakeland exactly once, and return to the square he was originally on. Each step, the knight moved to one of the squares nearest to that on which it stands but the queen can't reach.
The knight thought that since Cakeland has an even number of squares, he could achieve his goal this time.
However, the knight still failed.
Do you know why?
Hint
If you can't figure it out, try coloring Cakeland in another way:a d a d...
b c b c...
c b c b...
d a d a...
Then there are 12 squares of each color.
Answer
Color Cakeland as follows, i.e. each column alternatively abcd and dcba:a d a d...
b c b c...
c b c b...
d a d a...
Then there are 12 squares of each color. Again we suppose that the lower left corner is black. Then a and c are normal white squares while b and d are normal black squares. Additionally, we notice that each a-square must be preceded and followed by a b-square. This will require more b-squares than a-squares, but there are 12 squares of each color.
Therefore, the knight couldn't achieve his goal.
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Comments
Kiss, great teaser. I knew the knight wasn't able to do. I just couldn't have explained why like you did. Great job!
I got this one. It had to do with chess. Great one.
Great one Kiss, that was very well written keep these coming
Yes, maybe my secondary account writes better teasers than I do
Dec 13, 2007
Way too confusing. Who is the queen, and where can she reach? And how is this math related?
You explanation is better than what I came up with, but here it is anyway:
There are two internal squares that can reach each corner, so the knight must enter and exit from those squares to travel through the corner. Removing the two corners and the two entry and exit squares from one end of the chessboard leaves four more squares with exactly two squares that reach them. Two of the new entry-exit squares are internal. The pattern repeats until you get to the far end of the chessboard, where a conflict between the entry and exit squares for the side squares and the corner square occurs. There is no solution for any 4xN board.
(Sorry--the explanation is rather convoluted without drawing a diagram.)
There are two internal squares that can reach each corner, so the knight must enter and exit from those squares to travel through the corner. Removing the two corners and the two entry and exit squares from one end of the chessboard leaves four more squares with exactly two squares that reach them. Two of the new entry-exit squares are internal. The pattern repeats until you get to the far end of the chessboard, where a conflict between the entry and exit squares for the side squares and the corner square occurs. There is no solution for any 4xN board.
(Sorry--the explanation is rather convoluted without drawing a diagram.)
It never really explained that it was talking about chess, per se. It didn't say where the queen is. Impossible to comprehend.
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