Halp

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?

\(\begin{array}{|lclcl|} \hline \text{Let } d &-& \text{ down } &=& 0 \\ \text{Let } r &-& \text{ right } &=& 1 \\ \text{Let } u &-& \text{ up } &=& 2 \\ \text{Let } l &-& \text{ left } &=& 3 \\ \hline \end{array}\)

From M to "down-A", all distinct paths

are all three digit numbers to base 4:

\(\begin{array}{|r|r|l|} \hline base_4 &\text{path} & \text{spell the word MATH} \\ \hline 000 & ddd & \checkmark \\ 001 & ddr & \checkmark \\ 002 & ddu & \\ 003 & ddl & \checkmark \\ 010 & drd & \checkmark \\ 011 & drr & \checkmark \\ 012 & dru & \\ 013 & drl & \\ 020 & dud & \\ 021 & dur & \\ 022 & duu & \\ 023 & dul & \\ 030 & dld & \checkmark \\ 031 & dlr & \\ 032 & dlu & \\ 033 & dll & \checkmark \\ \hline && 7 \text{ distinct paths to spell the word MATH } \\ \hline \end{array} \)

For reasons of symmetry:

\(\begin{array}{|lcll|} \hline \text{From M to "down-A" there are } 7 \text{ distinct paths to spell the word MATH } & base_4 \quad 000 \to 033 \\ \text{From M to "right-A" there are } 7 \text{ distinct paths to spell the word MATH }& base_4 \quad 100 \to 133 \\ \text{From M to "up-A" there are 7 } \text{ distinct paths to spell the word MATH } & base_4 \quad 200 \to 233 \\ \text{From M to "left-A" there are } 7 \text{ distinct paths to spell the word MATH } & base_4 \quad 300 \to 333 \\ \hline \end{array} \)

\(4\times 7 = 28\) distinct paths can be followed to spell the word MATH