We've been trying to figure out this math for the last five days. It seems like no matter how hard we try, no matter if we work on it [day in and day out/day after day], we're not going to solve it.

1. Are there any mistakes?

2. Which is correct? What is the difference?
They're both correct, but both are normally used in describing a general (recurring) activity, rather than a specific one-time (single) task.

The first one focuses on the length of the days, the second one on the succession (number) of the days. That is, in "day after day," you might devote only a half hour to the task each day. "Day in and day out" is like "all day long," every day.

Day after day, I tell him to quit pestering me! (But maybe only once a day.)

Day in and day out, all I do is mop these damned floors!
Thanks, Avangi. Also, is the and in day in and day out optional?
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New2grammarThanks, Avangi. Also, is the and in day in and day out optional?

Technically, it should be there, but almost nobody says it that way. I'm not sure I've seen it in written form.
New2grammarThanks, Avangi. Also, is the and in day in and day out optional?
My experience comports with Philip's position, but I might not put it so extremely. I know a song, "Day In Day Out," and I know another song "Day In And Day Out." (I guess the first one is really "Thinking Of You.")

I know you don't believe in this, but Google leans the other way:
"day in and day out" two and a half million
"day in day out" one and a half million

- A.

Edit. In my first post, when objecting to your example, I should have used the word "habitual."

Both expressions are commonly used to describe habitual tasks, rather than one-time tasks.

As I revisit this, I realize that "Day after day he tried to solve the puzzle," or "Day after day he searched for his lost daughter," are both reasonable, where the tasks may or may not come to a logical conclusion. I guess 'day in [and] day out" would work here too.
Thank you, Philip and Avangi.
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If the guy works 9-5 and searches for his lost daughter, you wouldn't use 'day in day out' right? I just want to make sure.
DAY IN DAY OUT, he searched for his lost daughter
Good point, New2, I agree.