**Feb 2022**

The Monty Hall problem has now been a pox on humanity for two generations, diverting perfectly good brains away from productive uses.

So it is with great trepidation that I’d like to announce a new and more pernicious variant.

## Reminder

As a reminder, here’s the standard problem:

- There are 3 doors. A car is randomly placed behind one, and goats behind the other 2.
- You pick one door.
- Monty looks behind the other 2 doors. He chooses one of them with a goat behind it, and opens it.
- You get two options:
- Option A: You get whatever is behind the door you picked.
- Option B: You get whatever is behind the other closed door.

The well-known solution is that you should always switch, which gets you the car 2/3 of the time.

## Variant

In our variant, Monty is corrupt. The night before the game, you show up at his house with a suitcase of cash. He agrees that he will follow whatever algorithm you want when choosing the door to open. (Of course, he still can’t open a door with a car.)

What makes this challenging is:

- Monty doesn’t know where the car will be before the show starts.
- After the show starts, Monty can’t communicate with you other than by choosing what door to open.

The problem is: What is the best strategy for you and Monty to use? And if you use that strategy, how likely are you to get the car?

## Thoughts

The naive solution would be to play as if there was no conspiracy: If Monty picks a random door to open and you always switch, then you get the car 2/3 of the time. The question is: Can you do better?

I’m pretty sure I have the answer, but ~~people always yell at me when I talk about this stuff~~ I don’t yet have an elegant proof.

Do you have one? If so, please send it to me. When/if I get some nice solutions, I’ll post an update.

*Update*: I’ve gotten a few dozen proofs. Thanks everyone! I’m going to write an update with a summary of all the different proof strategies. If you have a particularly weird or elegant proof that you suspect other people haven’t thought of yet, please do send it, though!