The number of distinct paths to spell "MATH" on the grid, starting from the origin, is 3072. The paths follow a simple pattern based on the concentric squares around the origin, with specific counts for each letter's paths.
The paths for spelling "MATH" on the grid described was unnecessarily complicated and didn't directly address the specific structure of the grid. Here's a simpler approach:
1. Observe the pattern:
The key observation is that the letters are arranged in concentric squares around the origin:
- A at the vertices of the first square.
- T at the vertices of the second square.
- H at the vertices of the third square.
2. Count Paths for each letter:
- M (origin): There is only one starting point, so there is 1 path to reach M.
- A: At each A point, you can come from two directions (vertical or horizontal). Since there are four A points, there are 4 * 2 = 8 paths to reach any A point.
- T: Similarly, at each T point, you can come from two directions, and there are eight T points, so there are 8 * 2 = 16 paths to reach any T point.
- H: Again, at each H point, you can come from two directions, and there are twelve H points, so there are 12 * 2 = 24 paths to reach any H point.
3. Combine Paths:
To spell "MATH," you must go from M to A, then A to T, then T to H. Therefore, the total number of distinct paths is:
1 (M) * 8 (A) * 16 (T) * 24 (H) = 3072 paths.
Therefore, there are 3072 distinct paths to spell "MATH" on the grid starting from the origin.
Que. 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?