Here are the steps to solve this logic puzzle:
1. Albert knows the word immediately because he must have a unique letter - c, o, h, s, x, or i - that only appears in one word.
2. Bernard thinks for a moment, so he doesn't have a unique letter, but can eliminate words based on the unique letters Albert can't have.
3. Cheryl also thinks for a moment, so she doesn't have a unique letter either, but can determine the word with the process of elimination.
4. The only word left that is consistent with Albert, Bernard, and Cheryl's letters is "dog".
Therefore, the word they all knew is "dog