Problem: Sherlock’s Cubic

Sherlock Holmes is facing his nemesis Dr Moriarty in one final battle of the wits. Being a mathematician, Dr Moriarty has chosen their contest to be a challenge of integral calculus. Here are Moriarty’s rules:

1. Moriarty will secretly choose a cubic polynomial.
2. Holmes will do the same.
3. Each will integrate his own polynomial from -100 to 100.
4. Whoever obtains the lesser value will die at the hand of the other.

Holmes knows that Moriarty does not care who wins the integral contest—even if Moriarty dies by Sherlock’s hand, he will have won a pyrrhic victory by forcing Holmes to kill.

Holmes retorts “This challenge reduces to a child’s tease, ‘my father earns more money than your father’. Are we really battling over who can imagine the larger number? You will need to tell me more information about your cubic if you want me to take this seriously.”

“Very well Holmes. As a fifth rule, name three different x-coordinates, and I will tell you the y-coordinates as produced by my cubic. Then you may choose your own cubic. But three is all I will answer!”

“Agreed,” says Holmes. “Four points determine a cubic; that would be asking too much.” The battle proceeds as follows:

Moriarty chooses his secret polynomial $p(x)$. Holmes asks Moriarty for the values $p(-50)$, $p(0)$, and $p(50)$, and Moriarty tells him truthfully. Then Holmes chooses his own polynomial $q(x)$ after he hears Moriarty’s answers.

After the integration contest is over, Watson finally makes it to the scene. Both Holmes and Moriarty are alive.

“How did you manage that?” asks Watson.

“Elementary, Watson. I forced a draw. No one had to die if we both calculated the same value.”

To answer this puzzle, explain how Sherlock Holmes was able to pick a polynomial $q(x)$ which was guaranteed to tie with Moriarty’s $p(x)$ using only the values $p(-50)$, $p(0)$, and $p(50)$.