Annie is playing a game where she starts with a row of positive integers, written on a blackboard, each of which is a power of 2. On each turn, she can erase two adjacent numbers and replace them with a power of 2 that is greater than either of the erased numbers. This shortens the row of numbers, and she continues to take turns until only one number remains. Annie wins the game if the final remaining number is less than 4 times the sum of the original numbers. Is it always possible for Annie to win, regardless of the starting row of numbers?
then, by strong induction on n, we can reduce the first i numbers to a number x such that
x<4⋅2S=2S,
and we can reduce the last n−i numbers to a number y such that
y<4⋅2S=2S.
We now have x,y. WLOG, x≤y.
x,y⟶2y<4S.✓
There is not always an index i satisfying the condition considered in Hint 4. So we need to strengthen the induction hypothesis to find a suitable index i.Assume the numbers are b1,b2,…,bn.Instead of proving that the result is less than
4b1+4b2+⋯+4bn,
strengthen the statement and prove that the result can be less than or equal to
W=2b1+4b2+4b3+⋯+4bn−1+2bn.
Now we can find a suitable index i.
Let
Wr,s=2br+4br+1+⋯+4bs−1+2bs.
For a single number, define Wr,r=2br; for two adjacent numbers, Wr,r+1=2br+2br+1.If there is an index i such that