Disclaimer: i just fiddled with this, and it might not be exactly what you want. These two models will work, though I am sure there are better ways of doing it.
Here is some of the problems: if you need the checkerboard pattern to
be exactly as above (i.e. defined difference between black and white)
you cannot divide the shape into six. You must divide it into three.
If you need black-white checkerboard, keep 1/3, not 1/6, as going with 1/6 will give you this:
Trace your image (fiddle with the setting until you find the best solution):
ungroup. Then delete the parts you do not need:
Create two hexagons. Overlap them:
Delete all the lines that does not describe a diamond. This is "rotateable" by 120 degrees and will then describe a hexagon.
Then you could either redraw the checkerboard parts, or use this shape as a clipping mask over your existing jpg:
That would probably give you a more accurate version than suggestion #1.