Another complicating problem about The Lake House is that it is necessary in each iteration of history for certain major events to happen in much the same way for entirely different reasons. The biggest of these is that Kate must move into the lake house itself early in 2005. Alex thus must have left it. Then when Kate leaves the lake house, she must move into the Riviera Estates condominiums.
We have discussed the first two problems already. Alex left the lake house shortly after the death of his father; he reconciled to working with his brother, and moved closer to the office. Kate needed a place closer to her residency and hoped to live on the lake, so the property caught her fancy. Yet the final problem has its own problems.
Kate leaves the lake house shortly before Valentines Day 2006 to take a job in Chicago. She sends her forwarding address to the next tenant; it is delivered to Alex in 2004. From that moment Kate is tampering with history, and the tampering at times gets close to her own experience. Alex' father dies in her hospital; Alex rubs shoulders with her supervisor at that time. Alex plants a tree in front of the apartments which he would have planted at the house. He hides her book in the apartment she ultimately takes, tampering with the construction site before the apartments opened. He interacts with Morgan in ways that make the latter jealous and worried.
Perhaps none of this would prevent Kate from taking the lake house. The initial cause of her choice lay in events outside our knowledge, in the real estate agent pointing her in that direction. Nothing that Alex does necessarily prevents her from taking the house, and so we should allow the film the benefit of the doubt, that he leaves the house and she arrives, regardless of what other changes she makes to the past.
It is less clear that she would still get the job in Chicago, or that she would move to the Riviera Condominiums. It is entirely possible that in the original history she took a job in Madison, but that something she said to Alex in connection with the death of his father changed conditions in Chicago opening the job there. The original history could have had her elsewhere.
There are three ways this could have made a difference.
The biggest potential problem arises from the fact that she might not have been at Daley Square. Even had she taken a job at a different Chicago hospital her lunch with her mother might have been somewhere else in the city, and so the magic never would have been launched--she never kneels over his body trying to keep him alive wondering whether there is anyone in his life who will miss him. That would mean there was no time travel; the story never happened at all. Still, Daley Square is a likely place to meet for lunch if you aren't meeting at a restaurant, and the conjunction of schedules and weather means she might have met her mother here as long as she worked somewhere in Chicago.
The second potential problem here is that her supervisor would not have sent her to the lake house where she received the first letter. That could have happened if she'd had a different job, or if the supervisor had had a different job. Someone else might not have made the suggestion; it was not something she recognized herself.
Had she lived somewhere else, it might have impacted the story in other ways. Alex might not have found a construction site but an occupied residence; he would have planted the tree elsewhere. He would not have left the book in the floor of her room.
Yet none of this is impossible. Kate wanted to work in Chicago, and the job would have been open. The only complication is that the first letter she wrote contained her address, and the address changed. Even that is not a problem, as it simply means we have interlocking N-jumps: in the first version of the letter she gave a different address, but when history moved her to Riviera it happened before that first letter was written, and that timeline is repeated with the new address, triggering a repeat of all intervening timelines.
That's not so crazy as it sounds. Already with each new letter all previous anomalies repeat, as each letter is sent anew and each response is returned again as part of the history that already was. Wherever Kate winds up on Valentines Day 2006, that's where her first letter will say she was, and if in subsequent versions of time she moves the first letter will change such that it always gave the correct address.
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