Calculating Necessary Resistance In Current Source Circuit
You want to find the value of R3 that results in 1 A thru it.

Thevenin is your friend here.  Keep reducing the power sources to Thevenin equivalents, and combine them.  I1 and R1 are already a Norton source, so really easy to flip to Thevenin.  Then add R2 to that Thevenin source.  Now find the equivalent when that is connected to the Thevenin source of V3 and R4.  Now you have a single Thevenin source driving the R3,V1,V2 string.

That Thevenin voltage minus V1 and V2 is what is across RThevenin and R3.  By Ohm's law, find what RThevenin+R3 needs to be.  The subtract RThevenin, and the result is R3.