[smsm42]

They didn't just do it for sport, they tried to get it to surface and there aren't that many way to talk to a submarine that doesn't want to be found and is too deep to hear radio communications. They wanted the sub to surface, probably to enforce the blockade over Cuba. The captain of the sub erroneously decided that they're being attacked, because he had no communications with outside and didn't know what's going on. Arkhipov persuaded him not to rush and surface, and find out what's going on before firing.