### 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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