What is the fastest way to determine which element you are looking at while only using neutrons and nothing else?

It is not possible to positively identify an element based only on the number of neutrons.  Several different elements might even have the same number of neutrons.  For example, boron and carbon have 6 neutrons each.  Fluorine and neon each have 10 neutrons, so knowing neutron counts won't help a person positively identify the element.  

Additionally, different isotopes of the same element will have a different number of neutrons.  The element will have the same number of protons though.  

The only way to positively identify an element based on the number of subatomic particles is to know the number of protons present in the atom. The number of protons stays constant.  While looking at a periodic table, the atomic number provides a person with the number of protons.  The mass number tells a person how many protons and neutrons are present.  For example, the element fluorine has a mass number of 19 and an atomic number of 9.  Find the difference between the two numbers, and that is the number of neutrons.  19 - 9 = 10 neutrons.