Although characters from Half-Life 2 appear, the universe the mod starts out in differs slightly from the Half-Life universe; most notably the Combine and Rebels are friendly towards each other. Invaders attack the Combine and Rebels, requiring Gordon to stop them.
Mistake Of Pythagoras provides examples of the following tropes:
- Alternate Self: Everyone, but most notable with Dog. In this universe, he is implied to be a strider.
- Assimilation Plot: The dark invaders seem to want to convert the light invaders into them. The dark invaders are also seen building their structures on the Citadel.
- Attack of the 50 Foot Digits: The giant digits that fall from the sky.
- Brainwashed and Crazy: The implied reason for the existence of the hostile Combine in this universe.
- Colour-Coded for Your Convenience: Sort of. The dark invader infantry are more transparent than the friendly Combine.
- 11th-Hour Superpower: In the penultimate map, the white invader leader gives you an ability that surpases even the blue gravity gun from the original game: the ability to kill people and throw objects just by looking at them!.
- Establishing Character Moment: In the beginning, a citizen blatantly tells a Civil Protection officer - whose counterparts in the Half-Life universe are known for their Hair-Trigger Temper - to mind his own business. He complies immediately without question. Other CP units also won't try to hit the player if approached, and other Combine units help the player fight the invaders.
- Gravity Screw: One of the puzzles involves dropping objects into a hole until they stop in midair so the player can drop down safely. Anti-gravity is also present in the invader ship.
- Make Way for the New Villains: In the beginning, a strider is easily destroyed by one of the invader's vehicles.
- Mind over Matter: The light invaders give the player the ability to move objects simply by looking at them.
- Mook Maker: The dark invader ships shaped like a triangular prism can make more enemies.
- No-Sell: The dark invader triangular prism ships cannot be destroyed by small arms fire, but enough damage from rockets can disable one.
- Poor Communication Kills: The white invader leader helps the player fix the "mistake", but is shot dead by the friendly Combine. The Combine realize their mistake afterwards and bury the leader.
- Sky Surfing: The invaders use literal digits as elevators in their ship.
- Technicolor Fire: Anyone who is too close to the dark invader triangular prism ships will be burned by invisible fire.