Two math. wizards play in the following manner:

From a row of integers 0,1,2,…1023,1024 A erases 512 numbers
of his choice, - following this B erases 256 numbers of B’s choice.

Step 3: A erases 128 numbers, etc…

So at Step 10 player B chooses one of the 3 remaining numbers and erases it to define the amount of (dollars, pounds, euros, marbles) to be paid by A i.e. the difference between the two remaining numbers.

Clearly, A chooses a strategy to minimize this amount while

his opponent wants to maximize the outcome.

Assuming both follow the best strategy (Which?),

what will be the outcome of the game?

(In reply to

Maybe? by Omri)

I think Charlie has already shown that B's best strategy is not to take out the middle of any segment.

For instance, let's say that we get down to 9 contiguous numbers, 0-8. Then B (who has not used the best strategy up to now), removes 1,3,5 and 7. Now, no matter what 2 numbers A cross out, B will win at least 4.

If he uses his best strategy from the start, B can ensure winning 32.

(A's best strategy, of course, is not to play. For more detail on this strategy, see the move "War Games" starring a very young Matthew Broderick and Ally Sheedy.)