How can you lift an elephant with one hand?

Here is a question I've encountered:

How can you lift an elephant with one hand?

The answer provided is:

It is not a problem, since you will never find an elephant with one hand.

Now, if I actually want this sentence to mean that can I lift an elephant with a single hand (meaning my own hand), how would I say that?

The question is perfectly grammatical, but ambiguous (which is part of the joke, actually). Your own interpretation is just as valid as the one stated by the answer and makes more sense from a practical point of view.

It's possible to remove the ambiguity with one of the alternatives provided by the other answers but, really, you shouldn't.

One possibility is:

Using only one of your hands, how can you lift an elephant?

Even "How can you lift an elephant using only one hand" is ambiguous. The elephant may be using one of your hands.

"How can you lift an elephant with just one of your hands?" may be interpreted as an elephant that has stolen one of your hands, but not the other.

To be unambiguous, the sentence must be constructed such that the modifier is not adjacent to the object.

A possible re-writing is

How can you lift an elephant one-handed?

One-handed can act as either a adjective or an adverb. If it is placed after the object ("elephant"), the word order implies that it is being used in the adverbial sense, and so is modifying the verb ("lift") rather than the object. If you wanted to ask the "joke" sense of the question, you would instead say

How can you lift a one-handed elephant?

Answer:

Like this:

How can you lift an elephant one-handedly?

A definition of one-handedly from Oxford dictionaries:

Adverb

With or using only one hand.

Explanation of why it works ...

Some English grammar

There is a reason why this sentence—in contrast to the Original Poster's more intriguing one—is unambiguous.

In the original example, the preposition phrase with one hand could be modifying the verb phrase lift an elephant, or it could be modifying the noun elephant. One reason for this is that preposition phrases in English can modify verb phrases:

[[climb mountains] at the weekend]

And they can also modify nouns:

[[parties] at the weekend]

The example in this answer post, on the other hand, uses an adverb to modify the verb phrase lift an elephant. Adverbs can freely modify verb phrases:

It quickly evaporated
It evaporated quickly

But adverbs can't premodify nouns:

*It was a quickly evaporation (ungrammatical)

And they rarely postmodify them either:

*It was an evaporation quickly. (ungrammatical)

And that's why the adverb one-handedly can only be modifying the verb phrase lift an elephant and not the noun phrase elephant.

^{Notes for grammar junkies}

^{It used to be commonly thought that adverbs never modified nouns. However, recent work in corpus linguistics has shown this to not be true. Certain types of adverbs can very occasionally postmodify certain types of noun:}

^{- The riots recently are going to cause problem for years to come.}

^{Here we see recently modifying the noun riots. This can't be a sentence adverb, because the sentence as a whole is referring to the future, whereas recently refers to the recent past.}