The colloquial AmE expression “beats me” appears to be quite old:

[Greens Dictionary of Slang][1] 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? 


  [1]: https://greensdictofslang.com/entry/ic7gmxy#jbnh76a