Solution

Three horizontal paths intersect with three vertical paths to form a two by two square grid of paths with nine intersections. The top left corner of the grid has an entrance labelled Start, and the bottom right corner has an exit labelled Finish. Also, in the two top squares in the grid, there is a diagonal path joining the top right corner of the square to the bottom left corner of the square. In each of the bottom squares in the grid, there is a diagonal path joining the top left corner of the square to the bottom right corner of the square. The upwards direction is North.

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

Solution

We label 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. We also place arrows on the paths to show the direction in which Ezra can travel.

The three intersections on the top horizontal path are labelled S (left end), A (middle), and B (right end). The intersections on the middle horizontal path are C, D, and E, and the intersections on the bottom horizontal path are G, H, and F.

The arrows are as follows:

On the top horizontal path, there are arrows to the right: from S to A and from A to B.
On the middle horizontal path, there are arrows to the right: from C to D and from D to E.
On the bottom horizontal path, there are arrows to the right: from G to H and from H to F.
On the leftmost vertical path, there are arrows down: from S to C and from C to G.
On the middle vertical path, there are arrows down: from A to D and from D to H.
On the rightmost vertical path, there are arrows down: from B to E and from E to F.
On the top diagonal paths, there are arrows down and to the left: from A to C and from B to D.
On the bottom diagonal paths, there are arrows down and to the right: from C to H and from D to F.

We will count the number of routes from \(S\) to each intersection, and keep track of this information by placing a number at each intersection.

Since you can only travel south, east, southeast, or southwest along a path, from \(S\) there is only one route to \(A\) and only one route to \(B\). Therefore, we place a \(1\) at intersection \(A\) to keep track of the number of routes from \(S\) to \(A\). We also place a \(1\) at intersection \(B\) to keep track of the number of routes from \(S\) to \(B\).

To get to intersection \(C\), Ezra could travel directly from \(S\) or through \(A\). Since there is only \(1\) route from \(S\) to \(A\), there are \(2\) routes from \(S\) to \(C\). We place a \(2\) at intersection \(C\).

To get to \(D\), Ezra must travel from either \(A\), \(B\), or \(C\). Therefore, the total number of routes from \(S\) to \(D\) is equal to the number of routes from \(S\) to \(A\), plus the number of routes from \(S\) to \(B\), plus the number of routes from \(S\) to \(C\), or \(1+1+2=4\). We place a \(4\) at intersection \(D\).

To get to \(E\), Ezra must travel from either \(B\) or \(D\). Therefore, the total number of routes from \(S\) to \(E\) is equal to the number of routes from \(S\) to \(B\) plus the number of routes from \(S\) to \(D\), or \(1+ 4 = 5\). We place a \(5\) at intersection \(E\).

To get to intersection \(G\), Ezra must travel from \(C\). Therefore, there are only \(2\) routes from \(S\) to \(G\), and we place a \(2\) at \(G\).

To get to intersection \(H\), Ezra must travel from either \(C\), \(D\), or \(G\). Therefore, the total number of routes from \(S\) to \(H\) is equal to \(2+4+2=8\). We place an \(8\) at \(H\).

Finally, to get to intersection \(F\), Ezra must travel from either \(D\), \(E\), or \(H\). Therefore, the total number of routes from \(S\) to \(F\) is equal to \(4+5+8 =17\). We place a \(17\) at \(F\).

Therefore, there are a total of \(17\) different routes that Ezra can take from Start to Finish.

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