Question 9
A and $B$ play the following game with $N$ counters. $A$ divides the counters into $2$ piles, each with at least $2$ counters. Then $B$ divides each pile into $2$ piles, each with at least one counter. $B$ then takes $2$ piles according to a rule which both of them know, and $A$ takes the remaining $2$ piles. Both $A$ and $B$ make their choices in order to end up with as many counters as possible. There are $3$ possibilities for the rule:
$R1$ $B$ takes the biggest heap (or one of them if there is more than one) and the smallest heap (or one of them if there is more than one).
$R2$ $B$ takes the two middling heaps (the two heaps that $A$ would take under $R1$).
$R3$ $B$ has the choice of taking either the biggest and smallest, or the two middling heaps. For each rule, how many counters will $A$ get if both players play optimally?
$R1$ $B$ takes the biggest heap (or one of them if there is more than one) and the smallest heap (or one of them if there is more than one).
$R2$ $B$ takes the two middling heaps (the two heaps that $A$ would take under $R1$).
$R3$ $B$ has the choice of taking either the biggest and smallest, or the two middling heaps. For each rule, how many counters will $A$ get if both players play optimally?