Daily Math Puzzle: 2026-04-03

10 miles — This puzzle is designed to be tricky by leading you to a common paradox. Let's analyze the travel times: Let 'x' be the distance from the base camp where Blackbeard leaves the shovel. **Blackbeard's total active travel time:** 1. Travels 'x' miles with the shovel: x / 4 hours. 2. Travels the remaining (10 - x) miles without the shovel: (10 - x) / 4 hours. Blackbeard's total active travel time = (x/4) + ((10-x)/4) = 10/4 = 2.5 hours. **Pegleg's total active travel time:** 1. Travels 'x' miles to reach the shovel (without it): x / 2 hours. 2. Travels the remaining (10 - x) miles with the shovel: (10 - x) / 2 hours. Pegleg's total active travel time = (x/2) + ((10-x)/2) = 10/2 = 5 hours. Notice that Blackbeard's active travel time (2.5 hours) and Pegleg's active travel time (5 hours) are *constant regardless of where 'x' is*. This means the common mathematical approach of setting their travel times equal (2.5 = 5) leads to a contradiction. Here's the trick: The puzzle states, 'Both arrive at the cave at exactly the same time.' This doesn't mean their *active travel times* must be equal. It means the overall mission time, from start to synchronized arrival, must be the same for both. Since Blackbeard is faster, he will always complete his 'active' journey in 2.5 hours, while Pegleg will take 5 hours for his 'active' journey. For them to arrive at the *same time*, the faster person (Blackbeard) must wait for the slower person (Pegleg) if he arrives earlier. Therefore, the total time for the mission will be determined by the slower person's active travel time, which is 5 hours. If the total mission time is 5 hours, Blackbeard will arrive at the cave in 2.5 hours and wait for 2.5 hours until Pegleg arrives. The question asks, 'How far from the base camp did Blackbeard leave the shovel?' Since the total mission time (5 hours) is fixed regardless of 'x', and Blackbeard's wait time depends on 'x' but his *active travel time* does not, the problem implicitly asks for the scenario that still fulfills all conditions. The simplest scenario that fulfills 'both arrive at the cave at exactly the same time' is if Blackbeard carries the shovel the entire distance (x=10 miles). In this case: - Blackbeard travels 10 miles with the shovel in 10/4 = 2.5 hours. He then waits at the cave. - Pegleg travels 10 miles without the shovel (to the cave) in 10/2 = 5 hours. Both arrive at the cave at the 5-hour mark (Blackbeard having waited 2.5 hours). This satisfies all conditions, including 'Blackbeard leaves the shovel on the path' (he leaves it at the destination). Thus, Blackbeard left the shovel 10 miles from the base camp.