Daily Math Puzzle: 2026-08-25
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2026-08-25
In the magic show 'Time Traveler,' the magician has two assistants, Alice and Bob. Alice starts at one end of a 100-meter track and Bob at the other. At the same time, they both start walking towards each other. Alice walks at a speed of 4 meters per second, and Bob walks at a speed of 6 meters per second. However, every time they meet, they both turn around and start walking in the opposite direction, effectively making the track seem to 'expand' and 'contract' in a magical illusion. If they start walking at the same time and meet for the first time after 10 seconds, how many times will they have met by the time Alice has walked a total distance of 400 meters?
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Solution
6 times — To solve this, first calculate the combined speed of Alice and Bob when walking towards each other, which is 4 + 6 = 10 meters per second. In 10 seconds, they cover 10 * 10 = 100 meters, which is the length of the track, so they meet once. After the first meeting, Alice and Bob turn around. Now, they are walking away from each other. Alice's speed is 4 m/s, and Bob's speed is 6 m/s, so the track 'expands' at a rate of 4 + 6 = 10 m/s. Since Alice needs to walk 400 meters total and she walked 40 meters in the first 10 seconds (4 m/s * 10 s), she has 360 meters left to walk. Because the track 'expands' and 'contracts' in a pattern, each round trip (towards and then away) after the first meeting covers 100 meters of the track (50 meters towards and 50 meters away from each meeting point). However, Alice's speed is constant, so she covers 4 meters every second. For 360 meters at 4 m/s, Alice takes 360 / 4 = 90 seconds. Adding the initial 10 seconds, Alice walks for a total of 100 seconds. In these 100 seconds, considering the back-and-forth pattern and the speed at which they meet and turn around, they meet every 20 seconds after the first meeting because it takes 10 seconds to meet and then 10 seconds to be back at the starting points due to their combined speeds when moving towards and away from each other. From 10 seconds to 100 seconds, there are 4 intervals of 20 seconds (20s, 40s, 60s, 80s), meaning they meet 4 more times after the first meeting. Thus, including the first meeting, they meet a total of 5 times by the time Alice has walked 400 meters.
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