210209-2003 EST
cfg:
I think you have not had much training in electrical theory and troubleshooting. Sometimes problems can be simple to solve and other times very hard. When it seems hard, then it is good to follow a procedure that will hopefully lead you to a problem location.
I would start at the main panel. Here your voltages will change the smallest amount for load changes. I like to use a 1500 W spacer heater as a test load. At about 120 V this is a load change of about 12.5 A between ON and OFF.
At the main panel the voltage difference between the Neutral bus bar (common) (which is connected to the center tap of a 240 V center tapped single phase transformer, or neutral wire from a three phase transformer), and the EGC bus should be down in the millivolt range.
I would use the neutral bus at the main panel as my voltage reference point. If EGC to neutral voltage is small, there are no appreciable currents on EGC, and EGC is good to all locations, then I can use an EGC anywhere as a tested lead back to the main panel neutral. If this is not the case, then use an extension cord or a single piece of wire from neutral at the main panel as a long test lead to where you want to make a voltage measurement relative to the common of the main panel.
Suppose I get a voltage change at the main panel of 1 V between a hot phase conductor and the common bus for a 12.5 A load change on that phase, then I expect less change on the other phase. Possibly 1/2 as much. You won't see much of any light flicker in an incandescent with this 1 V voltage change at a nominal 120 V. I suspect that you won't find, at this amount of source impedance at the main panel, that there is much flicker at the main panel from the washer starting and stopping.
With background on the voltage change, at the main panel on the phase the washer is on, from starting and stopping of the washer, then look at the voltage change at the washer. This is possibly a number of volts.
Where your light is located see what the voltage change is from the washer cycling.
Next you look for what part of the light and washer circuits are common. Could be a neutral only or both neutral and hot. Use voltage measurements to narrow down where the commonality is. Difference in change in voltage between the washer and the light may give you an idea of where commonality exists.
Using EGC or a separate test lead wire to get a reference back to the main panel can help a lot in searching for commonality.
See where some measurements like these take you.
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