I'm hopeful that Quiz Kid Racer has similar ROM contents as patent US4051605. That patent looks like it's for the same calculator, but you could connect 2 of them together to compete, and the actual calculator doesn't have any connectors.
You can see the connectors on the right side of the calculators, as well as the cable. The patent says the units can be used separately in non-competition mode, which makes me think that the unit I bought without the connector likely uses the same ROM image; since you can't plug in a cable, it would always work in non-competition mode.