The colloquial AmE expression “beats me” appears to be quite old:
Greens Dictionary of SlangGreens Dictionary of Slang earliest usage dates from mid 19th c. But unluckily it doesn’t provide any suggestion about its origin.
1845 [US] W.T. Porter Big Bear of Arkansas (1847) 93: What on the Lord’s yearth young people now a days works and laces and befrils nite caps fur I can’t tell – it beets me.
Other online sources suggest a possible origin from the idea of being physically beaten.
Is there evidence of its more plausible origin?