King on Reduced Chessboard

Question

I've been set this as a task: On a chessboard, a king is to be allowed to move one square at a time: horizontally to the right, vertically downward, or diagonally to the right and downward. Imagine a reduced 4x4 chessboard, with the king beginning in the top-left square. By how many routes can he reach the bottom-right square? By how many routes can a similar journey be made on a full 8x8 chessboard?

I've searched on the web and it always comes up with matrixes. We haven't done any of those in class and my teacher will find out as we have to include our working out. Is there any other way of doing this? Or can someone help me understand matrixes?

Answer 1

HINT:

Try labeling each of the squares of the chessboard with the number of routes from the beginning square to that square:

The beginning square has just one route because the only way to get there is just to start there. The next square its right has only one route too because the only way is to move from the beginning one square to the right. The square below the beginning square has only one route as well because the only way to get there is to move one square down from the beginning.

But the square diagonally to the right and downward from the beginning square has three routes: to the right and down, down and to the right, or diagonally in one step.

So if you label the squares this way, the upper left hand corner will look like:

$$\begin{array}1 1&1\\1&3\end{array}$$

Now note that to get to any square in a single move, you have to have come from at most 3 others (which three?), and ask yourself: how is the number of routes to such-and-such a particular square related to the number of routes to the squares you could get to it from?

(By the way, I agree with you and JazzyMatrix that for the $4\times 4$ board, the right answer is 63.)

Answer 2

One way to introduce diagonals into this pathway problem is use the pascals triangle. I am assuming that you know what it is and if you don't, it's very simple to look up. The Pascal's triangle can be used to solve pathway problems when there are no diagonals. Imagine that the board is rotated and the number of pathways are labeled with each corresponding number on the pascals triangle. This will work for only horizontal and vertical movement and each number is the sum of the 2 above it:

However, when you add the use of diagonals, you will need to add the top 3 numbers to get the next number:

                                           1
                                         1   1
                                       1   3   1
                                     1   5   5   1
                                   1   7   13   7  1
                                 1   9   25  25  9   1
                               1   11  41  63  41  11  1

Ex. 63 = 25+25+13

As you can see, the answer 63 is in the answer. If you want to visualize the grid, you can tilt your head and see that a 4x4 grid starting on the top will be 63