Puzzle: Four Painter

Four painters need to paint a fence. Each painter has a different painting speed, and they take different amounts of time to paint the entire fence.

• Painter A can paint the entire fence in 1 hour.
• Painter B can paint the entire fence in 2 hours.
• Painter C can paint the entire fence in 4 hours.
• Painter D can paint the entire fence in 8 hours.

However, there’s a catch: Only two painters can work at a time on the fence. If they work together, they’ll complete the job in less time than if they work individually. Your goal is to find the shortest time it takes for all four painters to complete the job.

What is the minimum time it will take for all four painters to paint the fence, and which painters should work together?

Solution:

To minimize the time it takes to complete the fence, we should pair the fastest painter with the slowest painter, and the second fastest with the second slowest. This way, the combined painting rate of each pair is maximized.

Here’s how to pair the painters:

• Pair 1: Painter A (1 hour) and Painter D (8 hours)
• Pair 2: Painter B (2 hours) and Painter C (4 hours)

Now, let’s calculate the time it takes for each pair to complete the job:

• Pair 1 (A and D) completes the job in 1 * 8 / (1 + 8) = 8 / 9 hours.
• Pair 2 (B and C) completes the job in 2 * 4 / (2 + 4) = 8 / 6 hours, which simplifies to 4 / 3 hours.

Next, we need to determine how to combine these pairs to minimize the overall time:

Since the two pairs are working simultaneously, the total time will be determined by the longer of the two times. So, the minimum time it takes for all four painters to paint the fence is 8/9 hours.

Therefore, it will take 8/9 hours for all four painters to paint the fence. Painter A and Painter D should work together for the first part of the job, and Painter B and Painter C should work together for the second part. This minimizes the overall painting time.