Here is my third Fay With Your Math problem. Quick summary for newbies: these problems are offered to 3rd through 6th graders to solve for fun. Ideally they have a low entry point and a high ceiling. Candy is awarded to the best explanations of thinking and strategies. Apologies for the puns in the titles...they are definitely getting worse.This one is definitely emblematic of the types of problems I like to choose. It has a very low bar for entry -- anyone can find a few paths from A to B and start counting. But, it requires a lot of careful analysis to correctly find all of the paths. It is quite easy to double count or miss paths unless you have a strategy. There are many great problems like this. Looking at how many squares are on a chess board is a similar one where it is easy to count but hard to keep track of your counting.
I like that this problem has a high ceiling. While I don't expect many (if any) of the students to come up with a general formula for an n x m array, you could take this problem in that direction. As with other problems, I won't post a solution here. But, I will note that I was surprised to see Pascal's triangle show up in a common strategy for solving this.I think I should really stop being surprised when Pascal's triangle and triangular numbers show up in solutions. It happens so often that I should expect it more than I do. Does it show up in your strategy?
Here is a Printable PDF that has two copies per page.