Problem 1264

1. Total Number of Six-Letter Words:
The total number of six-letter words that can be formed using the letters A, M, and C (with repetition allowed) is given by:
$$3^6 = 729$$

2. Counting Words Containing the Substring "AMC":
We use complementary counting to find the number of words that do not contain the substring "AMC". First, we count the number of words that do contain the substring "AMC".

- Placement of "AMC":
The substring "AMC" can appear in the six-letter word in the following positions:
- Starting at position 1: AMCxxx
- Starting at position 2: xAMCxx
- Starting at position 3: xxAMCx
- Starting at position 4: xxxAMC

Thus, there are 4 possible positions for the substring "AMC".

- Remaining Letters:
After placing "AMC" in one of the positions, we have 3 remaining positions to fill with any of the letters A, M, or C. The number of ways to fill these 3 positions is:
$$3^3 = 27$$

- Total Count of Words Containing "AMC":
Therefore, the total number of six-letter words containing the substring "AMC" is:
$$4 \cdot 27 = 108$$

3. Subtracting Overcounted Cases:
We need to subtract the cases where "AMC" appears more than once, as they have been counted multiple times. Specifically, the word "AMCAMC" has been counted twice in the above calculation.

- Count of "AMCAMC":
There is exactly 1 such word: "AMCAMC".

- Adjusting the Count:
Therefore, we subtract 1 from our previous count:
$$108 - 1 = 107$$

4. Complementary Counting:
Finally, we subtract the number of words containing "AMC" from the total number of six-letter words to find the number of words that do not contain the substring "AMC":
$$729 - 107 = 622$$

Conclusion:
$$\boxed{622}$$