Professor Veryevil pointed the gun at Shirley Knowsalot. "And why shouldn't I just kill you now?" asked the professor.
"What about your shtick where you always give the hero a math puzzle that they can solve for their freedom?" responded Shirley.
"Oh yeah, I forgot about that. Let's see ... " The professor wished there was some sort of puzzling website he could call upon right now, but didn't know of one so he instead had to think fast. Looking around the room, and he eventually spotted a marker and a chessboard. "I challenge you to write an integer in every square of this chessboard so that
- the positive difference between any two squares that share an edge is at most $1$, and
- no integer appears more than seven times."
"What if I think that's impossible?" asked Shirley.
"Umm, well ... give me a proof of impossibility then. But it better be correct ... or else!" (awkward pause) "To clarify, the 'or else' means I'll shoot you with my gun."
"Could you be quiet and let me think?" responded Shirley.
How should Shirley respond to the professor's challenge?