To determine the maximum number of coins Morteza can collect regardless of how Joseph orders the boxes, we need to analyze the strategy Morteza can employ given the process of selecting boxes.

Key Points:

1. There are 100 boxes, each containing $i$ coins for $i = 1, 2, \ldots, 100$.
2. Morteza can pick the top 10 boxes in each of the 10 steps.
3. After opening the boxes, he can select any previously opened box to collect coins from it.

Strategy Analysis:

In the worst-case scenario, Joseph can arrange the boxes in such a way that the distribution of coins is not advantageous for Morteza. However, Morteza can maximize his collection by following a systematic approach:

1. Opening Strategy: In each of the 10 steps, Morteza will always open the top 10 boxes. Since there are only 100 boxes, over 10 steps, he will open 100 boxes.
2. Selecting Coins: Morteza needs to ensure that the boxes he chooses to collect from are maximized in terms of total coins.

Calculation:

To understand how many coins Morteza can guarantee, consider the following:

• In each step, he collects coins from one of the opened boxes. If he ensures that he is always selecting boxes with higher coin values over the rounds, he can maximize his collection.

Step-by-Step Breakdown:

• After the first step, he opens the top 10 boxes, which might contain any mix of the 1 to 100 coins.
• In subsequent steps, he should select boxes with the highest remaining values.

However, since he can only guarantee collection from 10 boxes at a time, it becomes clear that:

• He should try to ensure that he always has access to boxes that can give him higher amounts over the course of all 10 rounds.

Simulation of Worst Case:

To analyze the worst-case scenario:

• Assume that Joseph arranges the boxes such that each time Morteza opens boxes, he gets the lowest values available within the group of 10.

To derive a number:

• In each round, Morteza can at most collect the value of one box that he opens. Over 10 steps, the maximum sum of coins can be calculated as follows:

If the boxes are arranged such that he can access the maximum possible amounts systematically, he will gather:

• Guaranteed Collection: The ideal scenario would have him collecting:
  • First step: 10 (from boxes containing coins 1 to 10)
  • Second step: 20 (from boxes containing coins 11 to 20)
  • Third step: 30 (from boxes containing coins 21 to 30)
  • And so on...

Continuing this logic:

• The maximum he can gather would ultimately sum to collecting:

$\text{Coins collected} = 10 + 20 + 30 + ... + 100 = 10 \times (1 + 2 + 3 + ... + 10) = 10 \times \frac{10 \times 11}{2} = 10 \times 55 = 550$

Thus, Morteza can guarantee that he collects at least 550 coins regardless of Joseph's arrangement.

Conclusion:

The maximum number of coins that Morteza can always collect, regardless of the way Joseph orders the boxes, is 550 coins.

Would you like more details or have any questions about the solution? Here are some related questions to consider:

1. How would the strategy change if there were fewer boxes?
2. What if Morteza could pick more than one box per step?
3. How does the arrangement of boxes affect the outcome?
4. Can Morteza improve his strategy with different selection rules?
5. What is the significance of the coin distribution in this problem?

Tip: Always analyze the constraints of the problem carefully to determine the best strategy for optimization.