Aptitude Puzzle

On the right, you see a paper with a chessboard print on it. We want to cut the chessboard paper into pieces (over the lines!) such that each piece has twice as much squares of one color than of the other color (i.e. twice as much black squares as white squares or twice as much white squares as black squares).

Is it possible to do this? Give a proof!

Jacob and John are liars, who lie on specific days.

Jacob lies on Fridays, Saturdays and Sundays, but he is honest on all other days.

John lies on Tuesdays, Wednesdays and Thursdays, but he tells the truth on all other days.

On what day of the week would they both say “Tomorrow, I will lie.”?