Alice and Bob are ready to play a new game. Both the players take alternate turns. Alice starts first.

There are Nbinary numbers written on a blackboard.

Alice, in her turn, erases any 2 numbers from the blackboard and writes the bitwise OR of those 2 numbers on the blackboard.

Bob, in his turn, erases any 2 numbers from the blackboard and writes the bitwise AND of those 2 numbers on the blackboard.

Note that, after each move, the count of numbers written on the blackboard reduces by 1.

Both players play until a single number remains on the blackboard. Alice wants to maximise the remaining number while Bob wants to minimise the remaining number. Find the remaining number if both the players play optimally.

EXPLANATION:

Here we count the number of 1 and 0 written on the blackboard and store it in variables ones and zeroes respectively.
Now we observe that in one move Alice can take a 1 and 0 and remove the 0 from the board by taking bitwise OR while Bob can take a 0 and a 1 and remove the 1 by taking bitwise AND, provided both 1 and 0 exists.
In the end if only 1s exists then the last remaining would also be 1 only and same for 0.
Thus we just need to check the initial number of 1s and 0s and our answer would be based on whichever one is greater.

Bonus problem. I am not able to come up with solution:
what if instead of 0<=A[i]<=1 we have 0<=A[i]<=10^9 .
How do we get final answer?
Any help is appreciated!!

I feel (not able to prove it though) that u wud choose the largest and smallest number.

If u think about the numbers in binary format, then the largest number will have a 1 to the leftmost and the smallest number wud have a 1 to the rightmost. So if we take an OR, we are preserving all the 1s of the largest number and possibly add some from the smallest one. Similarly while taking AND, u r eating up the 1s of the largest number since the smallest contains 0s in those leftmost places. So that way we can always keep a set and keep picking the smallest and largest number for the operations.

I think this might work. But if this fails due to some other contradictory case, we will need to use DP i guess. Since DP is the only way to make this choice in a smart manner to always return the maximum/minimum result.

i didnt getting for equal number of ones and zeroes
for example A= 1 1 0 0
in first step, to maximize A= 1 1 0 ( xor of 1 and 0 = 1 reducing one zero)
in second step, to minimize A= 0 0 (to minimum xor of 1 and 1 will be taken and it was 0)
in thirs step , A = 0
finally answer was 0 but according to solution it was 1
Can anyone explain me where i am wrong?
thank you