CareerCup

Assuming the card array is not already sorted..

def buy_cards(cards, cash):
    if cards:
        cards = sorted(cards)
        cards = [0] + cards + [cards[-1] + cash]
    res = 0
    for n,m in zip(cards, cards[1:]):
        if not cash:
            break
        d = int((-2*n - 3 + ((2*n+3)**2 + 8*(cash-n-1))**0.5)//2)
        if d >= 0:
            d = min(d, m-n-2)
            res += d+1
            cost = (d*d + d*(2*n+3) + 2*(n+1))//2
            cash -= cost
    return res

print(buy_cards([2,3,5], 7))
print(buy_cards([2,3,5,11], 50))

Output:

2
7

Won't a greedy solution suffice ? You iterate from 1 to the amount D , and for each number you check whether your sister already has it, if she does not , you check whether you can buy the card ( you maintain the amount you spend ) , if yes then add 1 to the answer.

public class MaximumCardsToBuy {


    static int buyMaximumCards(List<Integer> sisCards, int amount) {
        if (amount == 0) {
            return 0;
        }
        int result = 0;
        int card = 1;
        while (amount > 0 && card <= amount) {
            if (!sisCards.contains(card) && amount >= card) {
                result++;
                amount -= card;
            }
            card++;
        }
        return result;
    }

    public static void main(String[] args) {
        List<Integer> sisCards = Arrays.asList(2, 3, 5);
        int amount = 7;
        System.out.println(buyMaximumCards(sisCards, amount));

        sisCards = Arrays.asList();
        amount = 10;
        System.out.println(buyMaximumCards(sisCards, amount));

        sisCards = Arrays.asList(2, 3, 5, 10);
        amount = 17;
        System.out.println(buyMaximumCards(sisCards, amount));
    }
}

public class MaximumCardsToBuy {


    static int buyMaximumCards(List<Integer> sisCards, int amount) {
        if (amount == 0) {
            return 0;
        }
        int result = 0;
        int card = 1;
        while (amount > 0 && card <= amount) {
            if (!sisCards.contains(card) && amount >= card) {
                result++;
                amount -= card;
            }
            card++;
        }
        return result;
    }

    public static void main(String[] args) {
        List<Integer> sisCards = Arrays.asList(2, 3, 5);
        int amount = 7;
        System.out.println(buyMaximumCards(sisCards, amount));

        sisCards = Arrays.asList();
        amount = 10;
        System.out.println(buyMaximumCards(sisCards, amount));

        sisCards = Arrays.asList(2, 3, 5, 10);
        amount = 17;
        System.out.println(buyMaximumCards(sisCards, amount));
    }
}

static int buyMaximumCards(List<Integer> sisCards, int amount) {
        if (amount == 0) {
            return 0;
        }
        int result = 0;
        int card = 1;
        while (amount > 0 && card <= amount) {
            if (!sisCards.contains(card) && amount >= card) {
                result++;
                amount -= card;
            }
            card++;
        }
        return result;
    }