Argument #1
The Court should rule in favor of Quarles. The Miranda decision is very clear concerning the requirement that suspects must be read their rights before questioning. And if they are not read their rights beforehand, anything that is said afterwards will not be admissible in court.
Argument #2
The Supreme Court should rule in New York's favor. The police acted properly when they asked the suspect where his gun was. The police were concerned with the safety of the public when they asked Quarles about the gun before reading him his rights. Therefore anything Quarles said can be used against him in court.