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When the day after tomorrow becomes yesterday,
the day which would then be today will be as far from Sunday as the day which is yesterday when it's two days after that day's tomorrow.

What day of the week was this spoken?

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The answer is

Wednesday

Explanation:

When the day after tomorrow becomes yesterday,

Three days from now (A)

the day which would then be today will be as far from Sunday as

the day which is yesterday when it's two days after that day's tomorrow.

Two days after that day.(B) A and B are two days apart and equally far from Sunday, so they must be Saturday and Monday, meaning the original day was Wednesday.

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This statement was spoken on

a Wednesday.
Day after tomorrow becomes yesterday on Saturday. And Saturday is 1 day away from Sunday.
Monday would be yesterday when it's Tuesday, which is two days after Sunday(which is Saturday's tomorrow).

And both Saturday and Monday are equidistant from Sunday.

An explanation to my strategy:

Consider our day of speaking to be Xday. We have two days of interest. First is that day on which day after tomorrow becomes yesterday which is the (X+3)th day. The second is yesterday when it is two days after (X+3)th day's tomorrow. Which is (X+3+1+2-1)th or (X+5)th day.

Writing down the week with Sunday as the centre, we can only choose Saturday and Monday as the days of interest. Hence, Wednesday has to be our Xday.

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