When Cortez arrived in Mexico in A.D.
Conclusion (thus)
The game originated sometimes between 1000 AD and Cortez' arrival in 1519 AD.
Evidence
The locals were playing the game, using a rubber ball, in 1519, and the local inhabitants began to use rubber around 1000.
Evaluate
It doesn't seem like we're being tested here on a Missing Link assumption, because the linking assumption would sound like,
Instead, we should probably try to argue the Anti-Conclusion.
Given that ... this game involved a rubber ball, and rubber wasn't used by these inhabitants of Mexico until 1000,
How could we argue that ... this game didn't originate between 1000 and 1519?
Is it possible the game originated before 1000 AD?
Sure. The local inhabitants didn't start using rubber until 1000, but maybe the earliest iterations of the game used a ball made of straw or some other material. Maybe around the year 1000, they switched to using a rubber ball, but the game was already invented before that.
It's also possible that the game originated in some other part of the world, where they began to use rubber earlier than 1000.
Goal
Look for an answer ruling out a possible objection. The correct answer, if negated, would potentially give us a way to argue that the game originated prior to the year 1000.