My 10 year old son is being given this class of question each week for homework but I'm a bit stumped on the best strategy to solve them.
A spider lies in ambush for the ant as shown below. How many different ways are there for the ant to reach home safely? Assume the ant can only move in the directions of → and ↑
My son's strategy is to run his pencil forward over all the paths he finds and sum them up. This leads to a mess and and under counting.
This is how I think I have solved it
I labelled the vertices with a grid and eliminated the illegal paths
Starting with vertex A4 and working down the column to E4, I added the number of possible paths from that point.
Working backwards, each prior vertex is the sum of its forward vertices. For example, the possible paths from B3 is (B4 + A3) = (2+1) = 3
Repeat until I get back to the start where E1 = (D1+E2) = 17+17 = 34
I think this is a directed graph problem. Is my way the best way to solve it for a 10 year old?