a precaution that gave him away.

[wowenglish1]

I wonder what precaution gave him away.

Any thief would empty the vase for water before stealing it to make it less heavy.
In order not to ruin his carpet and floor, the owner did that, but someplace else - a precaution that gave him away.

[JMurray]

In order not to ruin his carpet and floor, the owner did that, but someplace else - a precaution that gave him away.

precaution = something done in advance to prevent an unwanted/unpleasant result.
gave him away = showed him to be the culprit, the guilty person.

not a teacher

[5jj]

I wonder what precaution gave him away.

Why do you keep starting your threads with 'I wonder', when I have told you at least three times that 'I wonder'' is a comment, not a question?

[Raymott]

I wonder what precaution gave him away.

Any thief would empty the vase for water before stealing it to make it less heavy.
In order not to ruin his carpet and floor, the owner did that, but someplace else - a precaution that gave him away.

I agree that 'precaution' seems to be the wrong word. I'm not sure that I can identify what the precaution was either.
Also, the phrase should be "empty the vase of water".
Where did you find this puzzle? Is it probably written by a non-native speaker?

[JMurray]

@Raymott. You're right of course. On looking at this again it's clear that "precaution" is the wrong word. It's the owner's fastidiousness about his carpet that undermines the deception. Nothing to do with precaution.