The Kobayashi Maru is a scenario that is referenced in every installation of the Star Trek franchise. It is an infamous no-win situation that is part of the curriculum for command-track cadets at Starfleet Academy. The purpose of the test is to examine the character and command capabilities of cadets who are tasked with an impossible scenario. It aims to see how they manage under pressure when every available option results in failure, giving an idea of their discipline and poise.
The scenario originally put the cadet in control of the Enterprise in Gamma Hydra near the Klingon neutral zone. The ship gets a distress signal from the Kobayashi Maru, a disabled freighter within the zone. Entering the zone is a treaty violation; should the cadet choose to do so and save the freighter, they’re attacked by Klingon battle cruisers. Not entering the zone abandons the freighter, leaving everyone on board to die. There’s effectively no way to save the Kobayashi Maru and escape the zone without fight damage to the Enterprise.
Legend has it that Captain Kirk (William Shatner) is the only person to beat the scenario but did so only after reprogramming the computer with a winnable solution. The scenario is modified over time as relations with other races change for the better but ultimately maintains its unwinnable nature.
As the third-ranking officer on the Enterprise-D in Star Trek: The Next Generation (1987), Lieutenant Commander Data (Brent Spiner) should technically be required to take the test. He has aspirations of command, indicated in his conversation with Picard (Patrick Stewart) in Season 5, Episode 1, “Redemption: Part II.” He’s somewhat of an anomaly; by construction, he is a machine and does not have feelings or emotional responses regardless of the difficulty posed. His nature renders the core challenge of the Kobayashi Maru moot, but one can assume he still would have had to perform the exercise in training.
In Season 2, Episode 21, “Peak Performance,” Data plays a game of Strategema against a master player and concludes he has faulty programming; he puts himself on temporary leave and runs countless self-diagnostics to figure out how that is possible. Though not human, Data has his flaws. It would likely be very worth Starfleet’s time to examine the results of an android pitted against an unpassable test, since he's logically still bound to fail despite his superior intelligence and calculation skills. Would he loop infinitely? Would he shut down? Any number of things could happen. He has a lack of fear or empathy, but he’s just as vulnerable to the test for different reasons that no human test-taker would be able to replicate.