On the xy-plane, the origin is labeled with an M. The points (1,0), (-1,0), (0,1), and (0,-1) are labeled with A's. The points (2,0), (1,1), (0,2), (-1, 1), (-2, 0), (-1, -1), (0, -2), and (1, -1) are labeled with T's. The points (3,0), (2,1), (1,2), (0, 3), (-1, 2), (-2, 1), (-3, 0), (-2,-1), (-1,-2), (0, -3), (1, -2), and (2, -1) are labeled with H's. If you are only allowed to move up, down, left, and right, starting from the origin, how many distinct paths can be followed to spell the word MATH?

Hamburger Sep 14, 2024

#2**0 **

We are asked to determine how many distinct paths can be followed to spell the word "MATH" starting from the origin \( M \), given that movements are only allowed up, down, left, and right.

The points corresponding to \( A \), \( T \), and \( H \) are labeled on the xy-plane, and each of these labels corresponds to a specific set of coordinates.

### Step 1: Understanding the Problem

The problem provides:

- \( M \) is at the origin, \( (0, 0) \).

- \( A \)'s are at \( (1,0) \), \( (-1,0) \), \( (0,1) \), and \( (0,-1) \).

- \( T \)'s are at \( (2,0) \), \( (1,1) \), \( (0,2) \), \( (-1,1) \), \( (-2,0) \), \( (-1,-1) \), \( (0,-2) \), and \( (1,-1) \).

- \( H \)'s are at \( (3,0) \), \( (2,1) \), \( (1,2) \), \( (0,3) \), \( (-1,2) \), \( (-2,1) \), \( (-3,0) \), \( (-2,-1) \), \( (-1,-2) \), \( (0,-3) \), \( (1,-2) \), and \( (2,-1) \).

We need to determine how many distinct paths can be followed to spell "MATH", moving from \( M \) to an \( A \), then from \( A \) to a \( T \), and finally from \( T \) to an \( H \).

### Step 2: Movement Considerations

We are allowed to move only up, down, left, and right. This restricts the possible movements between the points labeled \( M \), \( A \), \( T \), and \( H \).

- From \( M \) at \( (0, 0) \), we can move to any of the \( A \)'s at \( (1,0) \), \( (-1,0) \), \( (0,1) \), or \( (0,-1) \).

- From each \( A \), we can move to one of the \( T \)'s that are one unit away from the \( A \)'s.

- From each \( T \), we can move to one of the \( H \)'s that are one unit away from the \( T \)'s.

### Step 3: Counting the Distinct Paths

Let’s break down the path counting process step by step.

#### Paths from \( M \) to \( A \):

From \( M = (0, 0) \), there are 4 possible \( A \)'s:

- \( A_1 = (1, 0) \)

- \( A_2 = (-1, 0) \)

- \( A_3 = (0, 1) \)

- \( A_4 = (0, -1) \)

So, there are 4 choices for the first step.

#### Paths from \( A \) to \( T \):

From each \( A \), we can move to a neighboring \( T \) that is one unit away. Let's examine the options for each \( A \):

- From \( A_1 = (1, 0) \), the possible \( T \)'s are \( (2, 0) \), \( (1, 1) \), and \( (1, -1) \). This gives 3 choices.

- From \( A_2 = (-1, 0) \), the possible \( T \)'s are \( (-2, 0) \), \( (-1, 1) \), and \( (-1, -1) \). This gives 3 choices.

- From \( A_3 = (0, 1) \), the possible \( T \)'s are \( (0, 2) \), \( (1, 1) \), and \( (-1, 1) \). This gives 3 choices.

- From \( A_4 = (0, -1) \), the possible \( T \)'s are \( (0, -2) \), \( (1, -1) \), and \( (-1, -1) \). This gives 3 choices.

Thus, for each \( A \), there are 3 possible \( T \)'s, so the total number of ways to move from \( A \) to \( T \) is \( 3 \times 4 = 12 \).

#### Paths from \( T \) to \( H \):

From each \( T \), we can move to a neighboring \( H \) that is one unit away. There are 3 neighboring \( H \)'s for each \( T \) (similarly to the calculation above). Therefore, for each \( T \), there are 3 possible \( H \)'s.

Thus, for each \( T \), there are 3 possible \( H \)'s, so the total number of ways to move from \( T \) to \( H \) is \( 3 \times 12 = 36 \).

### Step 4: Total Number of Paths

Multiplying the number of choices at each step, we get:

\[

4 \times 3 \times 3 = 36.

\]

Thus, the total number of distinct paths that can be followed to spell the word "MATH" is \( \boxed{36} \).

LiIIiam0216 Sep 14, 2024