Well, when you say "third number" do you mean the third 7? If so, your assumption of the linux permissions system is wrong.
There are 4 levels of permission:
0 - no permission
1 - execute
2 - write
4 - read
There are three different types of level of permission: owner, group, other.
The first number is the permissions the owner has on that file/folder, the second the group's permission, and the last anyone else's permission.
So 755 allows the owner to do anything, and group and others to read and execute, but not write.
So in your case, 666 should be sufficient in order to allow anyone to read and write to that directory/file.
Really, you should never have to have a file be 666. If a user uploads something, assuming they're using a php form, then the file would be saved as apache (or the user php is running as) and would then be the owner. At most, the folder the files are saved in should be 666, but no others. Only those that need to be 666 should be 666, all others should be more secure and more restrictive in permissions.