Weighing and deducing
I’m grappling with finding a defective ping-pong ball among 12 using a balance scale. The challenge is figuring out if the defective ball is heavier or lighter in three weighings.
OK, let me see. Evaluating 12 balls, with one defective, results in 24 scenarios. Using a balance scale in three weighings, I need to distinguish between these possibilities systematically.
Constructing examination
I’m detailing the method to conduct three weighings to identify the defective ball among 12, ensuring each scenario yields unique outcomes. Assigning outcomes to weighings effectively isolates the defective ball.
Mapping out weighings
I’m thinking through the standard solution, listing cases and devising three weighings to pinpoint the defective ball. Now, I’m setting up the table to connect outcomes to specific balls.
Weighing and narrowing
OK, let me see. In the first weighing, I split balls into groups and noted outcomes, narrowing down defective ball possibilities. The second and third weighings refine this further, focusing on specific groups.
Charting the course
OK, let me see. I’m working through the method for identifying a defective ball using three weighings, considering outcomes like balance, left pan heavier, and right pan heavier, and assigning letters to the balls for clarity.
Assigning code words
I'm assigning code words to each ball to define expected weighing outcomes, ensuring each ball's code word uniquely identifies it as defective (heavy or light).
Mapping out weighings
I’m organizing the results of three weighings to identify which ball, if heavy or light, causes the imbalance. Each weighing indicates if the left pan is heavier (L) or lighter (R) based on the presence of the defective ball.
Weighing analysis
I examined the outcomes of weighing various balls to determine their positions. For heavy ball 1, the code word is LLL. For heavy ball 2, the code word is L L R. Further analysis is needed to determine the code word for heavy ball 3.
Assigning code words
OK, let me see. I’m figuring out code words for balls in both heavy and light scenarios. For light balls in the Left pan, the Right pan goes down. Yeah, I’m working through the details for each ball now.
Mapping outcomes
I’m examining the outcomes for each ball in the left and right pans during weighings, identifying patterns in code words and defective balls.
Analyzing weighing outcomes
I’m examining the code words for each weighing to identify patterns in the ball heaviness and lightness. Balls have different code words for heavy and light states, indicating clear distinctions in their weights.
Noticing conflicts
I’m piecing together conflicts in code words for Ball 4 heavy and Ball 7 light. Adjusting weighings or sets of balls could ensure unique code words, making everything align nicely.
Mapping out the process
I’m crafting a strategy to identify a defective ball among 12 using a digital balance. The solution involves three weighings, each with specific ball groupings to narrow down the possibilities.
Determining the defective ball
I started by weighing 9, 10, 11, and 12, and depending on the balance, I identified the potential defective ball and whether it was heavy or light.
Breaking down possible outcomes
Starting with the left pan heavier scenario, I weigh different balls and analyze outcomes to pinpoint the defective one.
