This is the Base Model in Chemistry of Silver Nitrate and Copper.
2 AgNO3 (aq) + Cu (s) ---> Cu(NO3)2 (aq) + 2 Ag (s)
Assumptions in this model:
1- Each agent represent one atom or ion or ionic compound.
2- Each agent of water represents 1 molecule of water.
3- Every gray copper represent 1/2 oxidation process.
When copper atom reacts with a silver ion, it turns gray as an indication of a change in oxidation state from 0 to +1. Once another silver ion reacts with copper of oxidation state +1 ending with oxidation state +2, that Cu(II) ion dissociates into solution.
To be added to this model: Copper Nitrate flowing into solution and forming a complex with three water molecules turning the solution blue.
The count of copper within 200 steps counts the red copper atom and the gray copper (+1) ions. The Cu+2 ions are not counted there because they are actually a separate breed in this model. That make sense because Cu+2 is a product of the reaction, while Cu+1 is considered a transition state of the copper atom agent.
Note: The databox counting #copper atoms can be:
count red copper + (count gray copper/2) or
count all the copper agents, red and gray - (count gray copper/2)
When silver nitrates runs out before the copper atoms, then we might have two (or multiples of two) gray copper agents, which really should correspond to one Cu atom that didn't finish undergoing a full oxidation to Cu+2. so we have to divide the gray copper agent number by half to get a true number for copper atoms.
Another way to explain it:
The reaction:
2 AgNO3 (aq) + Cu (s) ---> Cu(NO3)2 (aq) + 2 Ag (s)
can be written as:
AgNO3 (aq) + 1/2 Cu (s) ---> 1/2 Cu(NO3)2 (aq) + Ag (s)
so this 1/2 ratio is why we divide the count of gray copper by half.
1/2 Cu is represented by one gray copper, which means 2 gray copper agents represent 1 Cu atom.
so to know how many copper atoms are left when all we have are gray copper agents, we take the gray copper count and divide by 2.