First, a little background. The "Kobayashi Maru" test is as follows:
A lone Federation starship is patrolling the Neutral Zone border. The ship receives a distress call from a transport ship - the Kobayashi Maru - from deep inside the Zone. The Maru should not be trespassing where it is, but claims to be under attack from the Klingons. A "winning" solution is to successfully rescue the Maru.
Obviously, one response is to ignore the distress call.
If the captain decides to help, the single Bird of Prey attacks the starship. If the starship defeats the single attacker, three BoP uncloak and attack. Should the Federation ship succeed against 3, then five more ships appear. This continues until the captain flees or the starship is destroyed. Various books have been written telling how the different characters have worked through this situation.
Kirk, however was able to (have someone) modify the scenario. When he attempted to rescue the Maru, and was challenged by the Klingons, upon announcing who he was, the Klingons retreated and allowed him to rescue the ship.
Kobayashi Maru from Wikipedia
As a side note, an unrelated but similar test is used to test the candidates. Called the "Psych Test", a variety of simulated emergencies take place in order to test the candidate's preparedness for being a starfleet cadet, such as Wesley's attempted rescue of two trapped scientists during a fire. TNG - Coming of Age This, while similar in forcing the person to make a tough decision in a no-win scenario, is not the KM, as it's used for Starfleet candidates, not cadets.