Starting with the A moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?

A

RRR

CCCCC

HHHHHHHH

We have one choice for A and three for R. After that, no matter which R we choose, we'll have 3 C's to choose from. After that, we'll have three H's to choose from. Therefore, the answer is 3^3 = 27. :D