Math on the Chessboard

Kings and horses

Problem 1

Let’s take a walk with a king from the upper left field of the chessboard (a8) to the lower right field (h1). How many different walk exists if your are stepping down or right only?

Solution:

Enter the number to each field how many different way we can get it from the left upper field. We can step to a field from the next above field or from the left next field (if they exist), so we can obtain this number for the actual field to sum the numbers from above and left fields. On this basis, if we fill in the fields as shown in the picture, we can obtain the result: 3432

Note: If anyone knows the meaning of binomial coefficients, it is easy to see that the

requested number is binomial coefficient , which is also:

Problem 2

Take a walk with a king over all of the fields of a chessboard starting from the upper left field (a8) and arriving in the lower right (h1) field. Every field can only go once. Find several possible walks.

Solution:

You will find some solutions on the next figures.

Problem 3

Take a walk with a horse over all of the fields of a chessboard starting from the upper left field (a8) and arriving in the lower right (h1) field. Every field can only go once. Find several possible walks.

Solution:

The two opposite field of chessboard have same color. The horse change the color of the field in each step. The horse starts from the upper left field and in the 63rd step arrived to the lower right field. It should be different color then the first field, but it is NOT. So horse can’t carry out the above conditions.