Arbitrary is "based on random choice or personal whim, rather than any reason or system". Ruling that a zombie can't trip a hydra isn't arbitrary, it's based on reasons of subjective plausibility.
It's based on the "reasoning" that the DM doesn't like imagining a zombie knocking a hydra prone in his own head.
But sure,
I'll play this game.
What is it that prevents the zombie from knocking down the hydra? The zombie's minion status? If minions can't knock down non-minions, then they shouldn't be able to knock down players, so clearly that's not it. The fact that the zombie is smaller than the hydra? If that's the case, no player power should be able to knock a large or larger creature prone. The fact that the hydra has multiple legs? If that's the case, how many legs does it take to gain utter immunity from being knocked prone?
Face it: this is an
unquestionably arbitrary decision based on the DM's personal conception of how he
feels a game mechanic should and should not work, to the point of ignoring the rules and the desires of his players.
I think the real problem is that you believe that the rules supersedes the DM's right to prioritize the narrative when it feels appropriate.
I believe that the desires of the DM often do not outweigh the desires of his players, and I believe this to be
especially true when the game's rules are on the player's side. And I've said this twice now.
You can leap to the defense of some imagined inviolate narrative all you want, but I
really doubt that allowing your players to knock a hydra prone with a minion would ruin
anything in your game, and
at most would require you to suspend the tiniest fragment of disbelief for a single monster's turn for the sake of the shared enjoyment your players will get out of their clever move.
There's one way to play chess.
Absolutely.
Oh, unless you count Fischer Random Chess. Or displacement chess. Or transcendental chess. Or upside-down chess. Or Dunsany's chess. Or handicap chess. Or the pawns game. Or peasant's revolt. Or Weak!. Or Active Chess. Or Alice chess. Or circular chess. Or Cubic Chess. Or cylinder chess. Or Chess Attack. Or Doublewide chess. Or flying chess. Or gravity chess. Or grid chess. Or hexagonal chess. Or infinite chess. Or Lord Loss chess. Or Los Alamos chess. Or Milennium chess. Or Minichess. Or Polgar Superstar Chess. Or singularity chess. Or three-dimensional chess. Or Absorption Chess (or Absorption Chess II). Or Accelerated Chess. Or Andernach chess. Or Antichess. Or Arimaa. Or Atomic chess. Or Benedict chess. Or Checkers chess. Or checkless chess. Or Chicken Chess. Or Circe chess. Or Crazyhouse. Or Einstein chess. Or extinction chess. Or Genesis Chess. Or guard chess. Or hierarchical chess. or Jedi Knight chess. Or knight relay chess. Or Knightmare. Or Legan chess. Or Madrasi chess. Or monochromatic chess. Or patrol chess. Or PlunderChess. Or refusal chess. Or replacement chess. Or rifle chess. Or Stationary King. Or Take-all. Or Three-check chess. Or ChessHeads. Or dark chess. Or dice chess. Or Fantasy Chess. Or Kriegspiel. Or No Stress Chess. Or Play It By Trust. Or Penultima. Or Schrodinger's chess. Or synchronous chess. Or avalance chess. Or doublemove chess. Or kung-fu chess. Or Marsellais chess. Or monster chess. Or progressive chess. Or zonal chess. Or Bosworth. Or bughouse chess. Or business chess. Or Djambi. Or Enochian chess. Or Forchess. Or four-handed chess. Or fortress chess. Or three player chess. Or Queen's Quadrille. Or Hippodrome. Or Anti-King chess. Or Baroque. Or Berolina chess. Or Bomberman chess. Or chess with different armies. Or Dragonchess. Or Duell. Or Gess. Or grasshopper chess. Or Maharajah and the Sepoys. Or omega chess. Or pocket mutation chess. Or Pole chess. Or Shako. Or stealth chess. Or 2000 A.D. Or Capablanca chess (or Capablanca random chess). Or Embassy Chess. Or Gothic chess. Or grand chess. Or Janus chess. Or modern chess. Or Seirawan chess. Or Chessers. Or Proteus. Or playing cards on a chess board.
Oh, and my personal favorite:
Knightmare Chess.
Credit to Wikipedia for that ridiculously long
list of chess variants.
How many different ways can you play D&D? Just one way?
Any number of ways. But this isn't about trying to find the One True Way to play D&D. This is advice, and it's good advice: if you put your personal desires above those of your players even when they have the rules on their side -
especially when it's over something as incredibly trivial as whether or not a zombie can knock a hydra prone! - you are sabotaging your own game.