Four explorers wish to get one of their number as far possible into the wilderness from their base
as
Each explorer can carry supplies for up to 10 days. At any time supplies can be transferred between
explorers and individual explorers can return to base, provided they have sufficient
supplies for the
return joumey. Supplies cannot be left unattended in the wilderness.
What is the greatest number of days the lead explorer can travel from the base so that all return safely?

This is a classic "explorer problem," where the goal is to get one person as far as possible into the wilderness while ensuring that all participants can return to base safely.

The key challenge here is managing the limited supply of food (each explorer can only carry supplies for 10 days) and optimizing the way supplies are transferred among explorers, allowing one of them to travel the furthest.

### Step-by-Step Breakdown:

1. **Understanding the Constraints**:
   - Each explorer can carry 10 days of supplies.
   - The lead explorer needs to go as far as possible, but everyone must return safely.
   - Supplies can be transferred between explorers, but they cannot be left unattended in the wilderness.

2. **Basic Strategy**:
   - Explorers will assist the lead explorer by accompanying them partway, transferring supplies, and then returning to base.
   - The key is to optimize when explorers turn back to maximize the distance the lead explorer can travel.

3. **Plan**:
   We can use the strategy of **gradual supply drops and withdrawals** to push the lead explorer farther. The idea is to stage supplies at specific points, so that explorers who drop supplies can turn back with enough food for the return journey while leaving some supplies for the lead explorer to continue.

Let's break this down into phases. We'll work with groups of explorers and split supplies to extend the range of one explorer as far as possible.

### Phase 1: 4 Explorers Travel Together
- All 4 explorers start with 10 days' worth of supplies (so, 40 total days of supplies).
- They travel together for 2 days, using up 8 days of total supplies (2 days per person). At this point, they will have 32 days' worth of supplies left (8 days each for 4 explorers).
- One explorer turns back, taking 2 days of supplies for the return trip, and leaving 6 days' worth of supplies for the remaining 3 explorers.

### Phase 2: 3 Explorers Continue
- The remaining 3 explorers continue traveling, now sharing the 6 days' worth of supplies from the first phase. They travel 2 more days, using up 6 more days of supplies.
- At the 4-day mark, there are 18 days of supplies left between the 3 explorers.
- One explorer turns back with 4 days of supplies (2 days to get back), leaving 12 days of supplies for the remaining 2 explorers.

### Phase 3: 2 Explorers Continue
- The 2 remaining explorers travel for 2 more days, using up 4 days of supplies (2 days per explorer). At the 6-day mark, they have 8 days of supplies left.
- One explorer turns back with 2 days of supplies, leaving 6 days of supplies for the lead explorer.

### Phase 4: The Lead Explorer Continues Alone
- Now, the lead explorer has 6 days of supplies and can continue for 3 more days into the wilderness before they must turn back (since they need 3 days to return).
- This explorer has now traveled a total of **9 days** away from base.

### Final Answer:
The greatest number of days the lead explorer can travel from the base is **9 days** into the wilderness, so the round trip would be 18 days in total (9 days out, 9 days back), with all explorers returning safely to the base.

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Would you like more details on how each phase works or on optimizing similar problems? 

Here are some questions to expand on this problem:
1. What happens if explorers can carry more than 10 days of supplies?
2. How would the strategy change if explorers could cache supplies in the wilderness?
3. What if one explorer was unable to carry the full 10 days of supplies?
4. How does the problem change if the return journey requires more supplies per day (e.g., due to difficult terrain)?
5. Could this strategy be applied in real-world situations like mountain expeditions?

**Tip**: When solving optimization problems, always try to break down the problem into smaller stages and look for repeating patterns or phases.

Four explorers wish to get one of their number as far possible into the wilderness from their base. Each explorer can carry supplies for up to 10 days. At any time supplies can be transferred between explorers and individual explorers can return to base, provided they have sufficient supplies for the return journey. Supplies cannot be left unattended in the wilderness. What is the greatest number of days the lead explorer can travel from the base so that all return safely?

Explore the classic explorer problem, where four explorers must maximize the distance one can travel into the wilderness while ensuring all return safely. This problem covers optimization, resource management, and logistical strategies, suitable for high school students. Learn how to distribute supplies to achieve the longest journey with a step-by-step breakdown.

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