I recognized that I had two different letters that needed to remain in the same place value of my answer; the M in the 512s place and the S in the 8s place.

I knew that those letters had corresponding numbers that needed to be either a zero or some combination of numbers in base 8 that would give me the same number; like 7 plus 7 (plus 1 as a carry over) yields 15, which is 17 base 8.

Base 8

M O S H

M O S H

_________________________________

A M U S E

**

7 5 0 3

+ 7 5 0 3

_________________________________

1 7 2 0 6

**

Solution

M = 7 H = 3 U = 2

O = 5 E = 6

S = 0 A = 1