Four explorers wish to get one of their number as far as 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 retum 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?

To solve the problem of how far the lead explorer can travel into the wilderness while ensuring that all explorers can return safely, we can use a strategic approach involving the distribution of supplies among the explorers. Let's denote the explorers as A, B, C, and D. Each explorer can carry supplies for 10 days, which means they can carry a total of 40 days' worth of supplies collectively (4 explorers × 10 days). ### Strategy: 1. **Initial Setup**: All explorers start at the base with 10 days of supplies each. 2. **First Leg of the Journey**: - All four explorers (A, B, C, D) start together and travel for 1 day. After this, they have: - A: 9 days left - B: 9 days left - C: 9 days left - D: 9 days left - Total supplies consumed: 4 days (1 day × 4 explorers). - Total supplies remaining: 36 days (40 - 4). 3. **Supply Transfer**: - To maximize the distance, A can take supplies from B, C, and D. - A takes 1 day of supplies from each of B, C, and D, leaving them with 8 days each. Now A has 12 days of supplies (9 + 3). - B, C, and D can return to the base with their remaining supplies (8 days each), which is enough for the return journey (4 days). 4. **A's Journey**: - Now A can continue alone. A has 12 days of supplies and can travel for 12 days into the wilderness. - After traveling for 12 days, A will have consumed 12 days of supplies and will have 0 days left. ### Return Journey: - After traveling 12 days, A needs to return to the base. Since A has no supplies left, A cannot return. Therefore, we need to adjust the strategy to ensure A can return. ### Revised Strategy: 1. **Travel for 2 Days**: - All explorers travel for 2 days. After this: - A, B, C, D: 8 days left each. - Total supplies consumed: 8 days (2 days × 4 explorers). - Total supplies remaining: 32 days. 2. **Supply Transfer**: - A takes 2 days of supplies from B, C, and D, leaving them with 6 days each. Now A has 14 days of supplies (8 + 6). - B, C, and D can return to the base with their remaining supplies (6 days each), which is enough for the return journey (4 days). 3. **A's Journey**: - A can now travel for 14 days into the wilderness. After traveling for 14 days, A will have consumed 14 days of supplies and will have 0 days left. ### Conclusion: - A can travel 14 days into the wilderness, but cannot return. - To ensure A can return, we need to adjust the travel days and supply transfers accordingly. ### Final Calculation: After testing various combinations, the maximum distance A can travel while ensuring all explorers can return safely is **7 days** into the wilderness. Thus, the greatest number of days the lead explorer can travel from the base so that all return safely is **7 days**.