Question 32
Gandalf the Wizard added to his arsenal of magic a new trick in which he simultaneously turns each integer into some integer different from it. Call an integer *a reflecting* if, for every integer $x$, the numbers $x$ and $a - x$ are turned into integers equal to each other. Is it possible that:
a. Numbers $1001$ and $1003$ are both reflecting;
b. Numbers $1000$, $1003$ and $1008$ are all reflecting;
c. Numbers $1002$, $1004$ and $1006$ are all reflecting?
a. Numbers $1001$ and $1003$ are both reflecting;
b. Numbers $1000$, $1003$ and $1008$ are all reflecting;
c. Numbers $1002$, $1004$ and $1006$ are all reflecting?