On a 10x10 "chessboard" a token is placed on the leftmost square of the bottom line a.k.a. a1.
You can move said token either one step to the right or one step up within the same column or one step diagonally (combining right and up).
To make it clear from c4 you may advance in one step to c5, c6 or d6.

How many distinct routes exist to reach the top line's rightmost square (i.e. j10?

I recognized this as a similar to another problem:

If you only allow right or up steps you generate Pascal's Triangle as the way to count the number of ways of getting to any given square.

Adding the extra move caused me to generate the triangle in a new way:  m(n,r) = m(n-1,r-1) + m(n-1,r) + m(n-2, r-1)
Where m(n,r) is analogous to C(n,r).

The central numbers form the sequence
1, 3, 13, 63, 321

Which, of course is in the OEIS:
https://oeis.org/A001850

The first comment there:
<table cellpadding="2" cellspacing="0"><tbody><tr><td width="20">
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Number of paths from (0,0) to (n,n) in an n X n grid using only steps North, Northeast and East (i.e., steps (1,0), (1,1), and (0,1)).

</td></tr></tbody></table>
And the solution is just the 10th term: 1462563

Edited on December 7, 2014, 8:30 pm

Posted by Jer on 2014-12-06 21:41:37