The question is how many paths exist from Start to Finish if you can move an arbitrary number of dots at a time up or right, but you cannot move down or left. The answer is 68 paths.
To solve this puzzle you could start by labeling each dot with a letter to keep track of your math, as shown below:
I started to solve the problem by drawing a tree diagram as I moved from letter to letter, and found there wasn't paper big enough to fit the entire tree! So, I came up with a modified tree diagram in which I wrote just the letters (no branches). Going through the lattice of dots for the most part you could move from one letter to two other letters, moving one dot at a time. A small piece of my diagram is shown below. In my diagram for any letter listed in a column I listed in the following column which letters I could move to. For instance, at A you could move either to B or to F, while at E you could move either to F or to J, as shown in the 3rd column.
| Start | A | B | C | ... |
| E | F | G | ... | |
| F | G | ... | ||
| J | K | ... | ||
| G | ... | |||
| K | ... | |||
| K | ... | |||
| O | ... | |||
| ... | ... | ... |
In my diagram I stopped everytime I reached either the top row or the rightmost column of the lattice, since once there the only way to go is either right or up, respectively, to the finish.
