Solving ZIO2015 Question 1: A Guide to Efficient Digit Selection

The problem of selecting digits in ZIO2015 Question 1 requires careful consideration of cost and selection criteria. While dynamic programming (DP) may seem tempting due to the nature of the problem, it can be quite tedious. Instead, a more efficient approach using a greedy algorithm will help you solve the problem more effectively.

Understanding the Problem

The question revolves around selecting a number with specific digit constraints and minimizing the total cost. The problem can be broken down into parts (a), (b), and (c), each requiring a different strategy to find the optimal solution.

Part (a)

In part (a), we need to find the largest number with at least one nonzero digit, given a certain budget. The basic idea is to calculate the maximum number of digits you can afford and then proceed in a greedy manner, always choosing the largest available digit without exceeding the budget.

For the maximum number of digits you can buy, you can afford 10 digits in total, with one of them being a non-zero digit. By selecting one '3' and nine '0's, you maximize the number of digits. Starting from the left, the first digit must be 3 or 7 since digits higher would exceed the budget. Therefore, you pick 7. With the remaining 28 rupees, the second digit can be 0 or 3, so you choose 3. The rest must be 0 to minimize the cost, leading to the answer 7300000000.

Part (b)

In part (b), the process is similar but with a different cost and budget. The cheapest digit available is '2' at a cost of 8. You can buy a maximum of 7 digits (48 rupees for the last six digits, leaving 14 for the first digit).

Choosing 5 for the first digit leaves 49 rupees. Since the remaining 6 digits (if all '2's) would cost 48, you need just enough for one more digit. The remaining 14 rupees can cover the remaining parts, ensuring the number is 5222222.

Part (c)

In part (c), the strategy involves picking the largest available digit within the budget constraints. The goal is to use up to 10 digits, with the highest total being 72 - 97 9. By selecting '9' for the first digit and '6' for the remaining nine digits, you ensure minimum cost and maximum usage of digits, leading to the number 9666666666.

Conclusion

Greedily selecting the most suitable digits within the constraints of the problem not only simplifies the process but also ensures an efficient solution. Understanding the problem's requirements and breaking it down into smaller, manageable parts can help in reaching the correct answer without unnecessary complexity.

By using a greedy approach, you ensure that you are always making the best choice at every step, leading to an optimal solution for each part of ZIO2015 Question 1.