BY HEART or OFFHAND or OFF HEAD?

English for Today by Ganiu Abisoye Bamgbose (Dr GAB)

She knows the poem ~off head~ (nonstandard).
She knows the poem offhand (nonstandard).
She knows the poem by heart (standard). If you know something such as a poem by heart, you have learned it so well that you can remember it without having to read it.

I could not give the figure ~off head~ (nonstandard).
I could not give the figure ~by heart~ (nonstandard).
I could not give the figure offhand (standard). If you do something offhand, you do it without premeditation or preparation.

Note that “off head” is not an idiom that relates to memory or memorisation. If you say that someone is off their head, you think that their ideas or behaviour are very strange, foolish, or dangerous:
I think he’s gone off his head. ✔️
GAB