[jonsen]

I pick 333 doors, and Monty opens 333 doors?

[kemotep]

No the variation with 1000 doors is that you open 1 door, Monty opens 998 doors and asks if you want to switch.

[AmericanChopper]

Yeah, the use of 3 doors, and the bit where one is opened just kinda hides what’s happening. Which is that you’re actually choosing between one random door, or ALL the other doors.

[scubbo]

The Eureka moment for me was similar, but slightly different. Consider a variant game with N doors. You pick a door, then Monty offers you two switch your payout to "the best single result from the remaining N-1 doors". Clearly, you would swap.

Now set N to 3, and consider that, by opening a door without the prize, that's exactly what Monty's offering you.