# Problem of the Week Problem C and Solution Corn Maze

## Problem

Baljit and Harinder go to a local farm to do a corn maze. The map of the corn maze is given.

On the day they arrive, the farm has the restrictions that they can only travel south, east, or southeast along a path. Using these restrictions, how many different routes can they take from Start to Finish?

## Solution

We can solve this problem by tracing out different routes and counting how many we find. We will set up a systematic approach to do so, to ensure that we do not miss any routes.

We begin by labelling the Start with the letter $$S$$ and the Finish with the letter $$F$$. We label the other seven intersections in the maze as $$A$$, $$B$$, $$C$$, $$D$$, $$E$$, $$G$$, and $$H$$, as shown.

Starting at $$S$$, Baljit and Harinder can only travel next to $$A$$ or $$C$$.

Case 1: Baljit and Harinder travel from $$S$$ to $$A$$.

Since Baljit and Harinder can only travel east, south, or southeast along a path, they have only two choices for where to go next: $$B$$ or $$D$$.

• If Baljit and Harinder travel to $$B$$, then since they can only travel east, south, or southeast, they must go to $$E$$ next, followed by $$F$$. Therefore, one route from $$S$$ to $$F$$ is from $$S$$ to $$A$$ to $$B$$ to $$E$$ to $$F$$.

• If Baljit and Harinder travel to $$D$$, then since they can only travel east, south, or southeast, they can go to $$E$$, $$F$$, or $$H$$ next.

• If they travel from $$D$$ to $$E$$, they must then go to $$F$$. Therefore, one route from $$S$$ to $$F$$ is from $$S$$ to $$A$$ to $$D$$ to $$E$$ to $$F$$.

• If they travel from $$D$$ to $$F$$, we have found another route. Therefore, one route from $$S$$ to $$F$$ is from $$S$$ to $$A$$ to $$D$$ to $$F$$.

• If they travel from $$D$$ to $$H$$, they must then go to $$F$$. Therefore, one route from $$S$$ to $$F$$ is from $$S$$ to $$A$$ to $$D$$ to $$H$$ to $$F$$.

In total, there are four routes from $$S$$ to $$F$$ in which Baljit and Harinder first travel from $$S$$ to $$A$$.

Case 2: Baljit and Harinder travel from $$S$$ to $$C$$.

Since Baljit and Harinder can travel east, south, or southeast along a path, they have three choices for where to go next: $$D$$, $$H$$, or $$G$$.

• If they travel from $$C$$ to $$D$$, they again have three choices for where to go next: $$E$$, $$F$$, or $$H$$.

• If they travel from $$D$$ to $$E$$, they must then go to $$F$$. Therefore, one route from $$S$$ to $$F$$ is from $$S$$ to $$C$$ to $$D$$ to $$E$$ to $$F$$.

• If they travel from $$D$$ to $$F$$, we have found another route. Therefore, one route from $$S$$ to $$F$$ is from $$S$$ to $$C$$ to $$D$$ to $$F$$.

• If they travel from $$D$$ to $$H$$, they must then go to $$F$$. Therefore, one route from $$S$$ to $$F$$ is from $$S$$ to $$C$$ to $$D$$ to $$H$$ to $$F$$.

• If they travel from $$C$$ to $$H$$, from $$H$$ they must go to $$F$$. Therefore, another route from $$S$$ to $$F$$ is from $$S$$ to $$C$$ to $$H$$ to $$F$$.

• If they travel from $$C$$ to $$G$$, they must then go to $$H$$ and then to $$F$$. Another route from $$S$$ to $$F$$ is from $$S$$ to $$C$$ to $$G$$ to $$H$$ to $$F$$.

In total, there are five routes from $$S$$ to $$F$$ in which Baljit and Harinder first travel from $$S$$ to $$C$$.

Therefore, there are a total of $$4+5 = 9$$ different routes that Baljit and Harinder can take from Start to Finish.