4. To celebrate Barbara's birthday, Alberto proposes the following game: given the set of numbers $0,1, \ldots, 1024$, Barbara removes $2^{9}$ numbers from this set. In the next step, Alberto removes $2^{8}$ numbers from the remaining ones. It's Barbara's turn again, and she removes $2^{7}$ numbers from the remaining ones, and so on, until only 2 numbers $a$ and $b$ are left. At this point, Alberto must pay Barbara $|a-b|$ euros.

Determine the maximum amount of euros that Barbara is sure to collect, regardless of the strategy adopted by Alberto.
Solution: The maximum amount Barbara is sure to collect is 32 euros.

At each move, Barbara can at least double the minimum distance between the remaining numbers. Indeed, on her first move, she can remove all the odd numbers, and on subsequent moves, regardless of Alberto's moves, she can remove the 2nd, 4th, 6th, ... of the remaining numbers in ascending order. After her 5 moves, the minimum distance between the remaining numbers will be at least \(2^5 = 32\).

On the other hand, Alberto can at least halve the maximum distance between the remaining numbers at each move. Indeed, after Barbara's first move, there will be \(2^{10} + 1 - 2^9 = 2^9 + 1\) numbers left; of these, necessarily either among those less than \(2^9\) or among those greater than \(2^9\), there will not be more than \(2^8\). Alberto can then remove the remaining ones, ensuring that only numbers in the interval \([0, 2^9]\) or only numbers in the interval \([2^9, 2^{10}]\) remain. Before Alberto's next move, \(2^9 + 2^8 + 2^7\) numbers will already have been removed, leaving \(2^7 + 1\) numbers, included in an interval of length \(2^9\). Consequently, either in the first half or in the second half of the remaining interval (excluding the central number), there will not be more than \(2^6\), and therefore Alberto can remove them all, leaving only numbers in an interval of length \(2^8\). Continuing with this strategy until his fifth and final move, Alberto can ensure that only two numbers remain in an interval of length \(2^5 = 32\), and thus with a distance not exceeding 32.

In conclusion, Barbara is sure to earn at least 32 euros, Alberto is sure not to have to pay more than 32 euros, and therefore 32 euros is the maximum amount Barbara is sure to collect.