These three men quarrelled ______ themselves.

Concept

The preposition that expresses a reciprocal relationship distributed across a group of three or more people is "among". "Between" expresses a relationship involving only two parties (or two clearly separate parties considered one-to-one).

Application

The sentence refers to "these three men" — a group of more than two. A quarrel shared reciprocally within that group of three is therefore expressed with "among", giving "quarrelled among themselves".

Contrast

• "between" — fits a two-party relationship (e.g. between two men), so it does not match a group of three.

• "with" — needs a separate object the men quarrel against (quarrel with someone), not a reciprocal action turned inward on "themselves".

• "against" — marks opposition to an external party, again requiring an outside opponent rather than a mutual quarrel inside the group.

Hence the blank takes "among": quarrelled among themselves.