Pushing scholarship credited in December '21 for Spring '22 bill into 2022 tax year

Generally, what happened in 2021 stays in 2021 and is reported in 2021. 

What happens in 2022 stays in 2022  and is reported in 2022.

If you got paid in 2021 it is reported in 2021.  Thus, if the 2021 payment was made to you, then you report receiving it in 2021. 

If the payment was made in 2021 to your school account for your 2022 education, then the amount paid could be eligible for the education deduction or credit in 2021.

If the payment was made to you in 2021, and you made the payment to the school in 2022, then the 2021 payment received is reported in 2021, and the 2022 payment to the school is reported in 2022.

No, it is not possible.  Moreover, given that your scholarship was applied to room and board and not a qualified educational expense, such scholarship is taxable.   In order for a scholarship to be tax free, the IRS has stated, in relevant part, that it must be of the following type:

"A scholarship or fellowship grant is tax free only to the extent:

• It doesn't exceed your qualified education expenses;
• It isn't designated or earmarked for other purposes (such as room and board), and doesn't require (by its terms) that it can't be used for qualified education expenses; and..."

Tax Benefits for Education

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