public int lastRemaining(int n) {
boolean left = true;
int remaining = n;
int step = 1;
int head = 1;
while (remaining > 1) {
if (left  remaining % 2 ==1) {
head = head + step;
}
remaining = remaining / 2;
step = step * 2;
left = !left;
}
return head;
}
My idea is to update and record head in each turn. when the total number becomes 1, head is the only number left.
When will head be updated?
 if we move from left
 if we move from right and the total remaining number % 2 == 1
like 2 4 6 8 10, we move from 10, we will take out 10, 6 and 2, head is deleted and move to 4
like 2 4 6 8 10 12, we move from 12, we will take out 12, 8, 4, head is still remaining 2
then we find a rule to update our head.
example:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Let us start with head = 1, left = true, step = 1 (times 2 each turn), remaining = n(24)

we first move from left, we definitely need to move head to next position. (head = head + step)
So after first loop we will have:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  > 2 4 6 8 10 12 14 16 18 20 22 24
head = 2, left = false, step = 1 * 2 = 2, remaining = remaining / 2 = 12 
second loop, we move from right, in what situation we need to move head?
only if the remaining % 2 == 1, in this case we have 12 % 2 == 0, we don't touch head.
so after this second loop we will have:
2 4 6 8 10 12 14 16 18 20 22 24  > 2 6 10 14 18 22
head = 2, left = true, step = 2 * 2 = 4, remaining = remaining / 2 = 6 
third loop, we move from left, move head to next position
after third loop we will have:
2 6 10 14 18 22  > 6 14 22
head = 6, left = false, step = 4 * 2 = 8, remaining = remaining / 2 = 3 
fourth loop, we move from right, NOTICE HERE:
we have remaining(3) % 2 == 1, so we know we need to move head to next position
after this loop, we will have
6 14 22  > 14
head = 14, left = true, step = 8 * 2 = 16, remaining = remaining / 2 = 1 
while loop end, return head