Somehow my solution became the top rated for a time and I have no idea when that happened. The last time I checked it was like the fifth one, so apparently a bunch of other solutions were culled by the site or deleted by their owners.
Now under a normal situation, you would boot the game once a week for 12 consecutive weeks. Obviously this is a crock so everyone has been trying really hard to figure out where this game delimits one week from the next, so that someone could just swap their timezone back and forth to complete this achievement in a single day.
The universal conclusion on how to determine the day and time you need to do this on: "fuck if I know". Whatever date methods the developer is using to determine week starts/ends is arbitrary and changes every year. Making matters worse is that a day that works in one country tends not to in another, without any obvious correlation between them like time zones. There is no way for anyone to know without simply busting their ass to experiment on their own.
So, the best we've got is for some poor bastards to figure out what the current year's magical day is on their own and hopefully share their results with the rest of the class in the comments. Your only remaining option is, as HolyHalfDead puts it, "betting on every horse in the race" by starting the game 7 or more times per day for a week, though anyone who can grasp odds and sports betting knows that you would lose a spectacular amount of money this way (or in this case, time.)
In the end, I'd say just bite the bullet and boot the game once a day for three weeks. Even the methods that are supposed to be rock solid and work 100% of the time are rigmaroles (I mean, just take a glance at any other solution for this achievement) that do not work on certain days, at certain times, when certain planets are aligned with each other, or when you're not standing on your head with a strawberry ring pop stuck in your ass.