The following moves give a final score of 9: [1, 5, 2, 1].

I'm trying to figure out n = 12 and this is how to get 9 and 3

n = 12
Alice Turn: Alice = 1, Bob = 0, Remain = 12 - 1= 11
Bob Turn: Alice = 1, Bob = 1, Remain = 12 - 2 = 10
Alice Turn: Alice = 5 + 1 = 6, Bob = 1, Remain = 12 - 7 = 5
Bob Turn: Alice = 6, Bob = 1 + 1 = 2, Remain = 12 - 8 = 4
Alice Turn: Alice = 6 + 1 = 7, Bob = 2, Remain = 12 - 9 = 3
Bob Turn: Alice = 8, Bob = 2 + 1 = 3, Remain = 1
Alice Turn: Alice = 8 + 1 = 9, Bob = 3, Remain = 0

Try to play the game on a piece of paper and try different strategies and try to find the best one.

I don't understand this.
12 is an even number so why is only 1 cards taken first?

12,11,10,5,4,2,1

alice -9
bob -3

Tell me someone. Why is the result 8 for 12 wrong?