I want to know how many puts I have to buy at a predefined level so that if I average down to that level on a big loss I will recover some/most of the loss.

That's not something that math can solve because how exactly the puts are going to cover your losses is also going to depend on demand & supply & liquidity of the puts at the time of the loss, something that cannot be calculate with 100% certainty. There are various pricing models that can estimate the future price of puts but they are not exact. So if you want to make sure your potential loss is well-covered by the Option, my suggestion is buy the option with strike price close to the entry price of the underlying. So that way, in case if the liquidity is bad and you can't sell the option with enough to cover the loss, at least you can exercise the option and won't lose much. But be prepared to pay because it's going to be EXPENSIVE!!

The idea is to buy ES on the way down but initiate the position with a bunch of puts a lot lower in case it keeps going down and down.

Well, delta will give you the instantaneous ratio of shares to contacts, and gamma (I think...whichever one is delta delta) will tell you how that changes...but that ratio will change with the price in the underlying. But what you're looking to do is just trading dollars and paying the spread and commissions on top of that. It's a losing to delta hedge your positions...in fact, that's where options market makers make money (and you're taking about fading that trade)

Assuming that the underlying always lands in profit of course. If the underlying actually ends up in a loss, the option buying would be all worth it and the MM's will be screwed especially if they get exercised upon. But yeah during all those times that the underlying actually lands into profit, Market-making in Options is PURE JOY!!!

Market makers are fully hedged. Their goal is to transfer that risk elsewhere after selling you the option be it with either already existing option inventory that offsets the position or via the underlying itself (delta hedging).

A market maker tries to have no position in the underlying... so they will hedge soft delta's with soft delta's... (options with options). Having the underlying on your book is a cash/capital drain for MM's, so it's better to have no stocks... In the end you want boxes.

Exactly, so to get into a profit from a losing underlying position.. you would need more (short) delta's in puts than long delta's in underlying. The exact number of puts depends on 1. the initial size of underlying and level bought... 2. which put you would buy.... 3. the time to expiry of those puts.... and 4. the implied volatility of those puts.... 5. how fast you want that loss to turn to break-even/profit.. (basically these all determine the price of the option)....

To do this https://www.elitetrader.com/et/threads/martingale-system-in-trading.146187/page-2#post-2196097