# On 8x8 board, a knight is on the second square of the last row. Only moving upwards, how many routes to the top?

In chess, a knight moves in L-shaped jumps consisting of two square along a row or column plus one square at a right angle. It can only move upwards. How many routes can it take to any square on the top row of the board?

• So the knight starts where it usually starts, and you want the number of ways it can reach the top row? – greenturtle3141 Sep 25 '16 at 23:47
• Yes, the knight starts at the second square of the bottom row and all the ways it can reach the top row. Thanks. – wohc notna Sep 25 '16 at 23:55

The easiest way to do this, I think, is

just to fill in the number of routes from the top row (where the number is already 1: you're on the top row and there's only one way to get to the top row, namely by making no moves) downward: at each stage, you put in each square the sum of the numbers in the squares you can move to from there. It's quicker and more reliable to do it with a computer, but it's well within the range of reasonable hand calculation.

 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 3 4 5 5 5 5 4 3 6 8 11 13 13 11 8 6 15 21 28 29 29 28 21 15 36 46 65 73 73 65 46 36 86 116 159 168 168 159 116 86 205 269 373 413 413 373 269 205