91Ó°ÊÓ

Murray's brother. On another archeological dig, Alley discovered another ancient oil lamp. Again she rubbed the lamp, and a different genie named Curray appeared. After Alley explained her run-in with Murray, Curray responded, "Well, since you know my brother Murray, it's like we're almost family. I'm going to give you four wishes instead of three. What do you say?" Since things had worked out so well the last time, she said, "I already found the Rama Nujan, so now I'd like to find the Dormant Diamond." "You got it," replied Curray. And instantly 12 identical-looking stones appeared. She then used her last three wishes to acquire three balance scales. Each scale was clearly labeled, "One Use Only." Alley looked at the stones and was unable to differentiate any one from the others. Curray explained, "The diamond is embedded in one of the stones. Eleven of the stones weigh the same, but the stone containing the jewel weighs either slightly more or slightly less than the others. I am not telling you which - you must find the right stone and tell me whether it is heavier orl ighter." Alley could use each of the three balance scales exactly once. She was able to select the stone containing the Dormant Diamond from among the 12 identical- looking stones and determine whether it was heavier or lighter than each of the 11 other stones. This puzzle is a challenge. Try to figure out how Alley might have accomplished this feat. Further Challenges

Step by step solution

01

First Weighing

Divide the 12 stones into three groups of 4 stones each. Let's label them as groups A, B, and C. Use the first balance scale to weigh Group A against Group B.

02

Analyze First Weighing Result

If Group A equals Group B, the Dormant Diamond must be in Group C. If Group A is not equal to Group B, the Dormant Diamond is in the group that weighs differently.

03

Second Weighing

Assume from Step 2 that the Dormant Diamond might be in either Group C or in the heavier/lighter group (if Groups A and B were unequal). Take the stones from the suspicious group (either A, B, or C depending on earlier results) and divide into three groups of 1, 1, and 2 stones. Weigh the group of 1 stone against the other group of 1 stone.

04

Analyze Second Weighing Result

If the group of 1 stone equals the other, the Dormant Diamond is among the two stones not yet weighed. If they are not equal, the heavier or lighter stone is the suspect.

05

Third Weighing

Using the remaining or known two stones from Step 4, weigh them on the balance scale. Weigh one stone against the other to determine which is heavier or lighter. This will allow you to identify which stone contains the diamond.

06

Final Verification

Based on the outcome of the third weighing, precisely identify which stone is heavier or lighter.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Problem Solving

In the context of this mathematical puzzle with Alley and the 12 stones, problem solving is all about breaking down the challenge into manageable steps. Problem solving involves:

• Identifying and understanding the specific problem posed: finding the unique stone with the Dormant Diamond, and determining whether it is heavier or lighter.
• Analyzing the available resources: in this case, Alley has three balance scales, each with a single use.
• Planning a strategy: dividing stones into groups and conducting a systematic weighing process.

The key to problem solving in this exercise is optimization—using the least number of steps to reach a solution. This often involves creative thinking and pattern recognition. To apply problem solving effectively here, Alley logically divides stones into smaller groups and narrows down the possibilities with each weighing. With a systematic approach, it's easier to manage the complexity of the problem, reducing the set of possibilities each time and making strides toward the final solution.

Logic

Logic plays a central role in solving the puzzle with the stones because it involves reasoning and forming conclusions based on available information. In our case:

• Alley uses deductive reasoning to draw conclusions from the outcomes of each weighing.
• Logic dictates each choice of stone groups to weigh, anticipating all possible outcomes and their implications.
• She uses an 'if-then' approach to determine her next steps after each weighing, depending on whether groups of stones balance or one is heavier/lighter.

Understanding logical principles helps Alley in determining which stones to leave out and which ones need further weighing. In essence, logic helps to establish a clear sequence of actions contingent upon previous outcomes, ensuring that each move is informed and purposeful. For example, by deducing that if two groups balance, the Dormant Diamond is in the third group, Alley systematically narrows down her search, enhancing efficiency and accuracy in her approach.

Critical Thinking

Critical thinking in this scenario is about evaluating the situation and making reasoned judgments to solve the puzzle effectively. It involves questioning assumptions and analyzing patterns:

• Alley needs to critically assess each outcome from the balance scale and decide her subsequent moves.
• She must consider various possibilities: Does the stone weigh more or less and which stones should she weigh next?
• By constantly reassessing and adapting her strategy based on new insights from weighings, she exercises adaptive thinking.

Critical thinking allows Alley to adjust her strategy dynamically and make informed decisions. She continuously filters out irrelevant information and focuses on specific clues given by the weighings. It empowers her to differentiate between crucial and non-crucial information and to ask the right question at each step: "What does this result mean, and how can it guide my next weighing?" Such rigorous evaluation ensures that Alley's approach is not only methodical but also flexible enough to encompass all potential outcomes.