Math Problem Statement
On an archeological dig near the highlands of Tibet, Alley discovered an ancient oil lamp. Just for laughs she rubbed the lamp. She quickly stopped laughing when a huge puff of magenta smoke spouted from the lamp, and an ornery genie named Murray appeared. Murray, looking at the stunned Alley, exclaimed, "Well, what are you staring at? Okay, okay, you've found me; you get your three wishes. So, what will they be?" Alley, although in shock, realized she had an incredible opportunity. Thinking quickly, she said, "I'd like to find the Rama Nujan, the jewel that was first discovered by Hardy the High Lama." "You got it," replied Murray, and instantly nine identical-looking stones appeared. Alley looked at the stones and was unable to differentiate any one from the others.
Finally she said to Murray, "So where is the Rama Nujan?" Murray explained, "It is embedded in one of these stones. You said you wished to find it. So now you get to find it. Oh, by the way, you may take only one of the stones with you, so choose wisely!" "But they look identical to me. Howill I know which one has the Rama Nujan in it?" Alley questioned. "Well, eight of the stones weigh the same, but the stone containing the jewel weighs slightly more than the others," Murray responded with a devilish grin.
Alley, becoming annoyed, whispered
under her breath, "Gee, I wish I had a balance scale." Suddenly a balance scale appeared. "That was wish two!" declared Murray. "Hey, that's not fair!" Alley cried. "You want to talk fair? You think it's fair to be locked in a lamp for 1729 years? You know you can't get cable TV in there, and there's no room for a satellite dish! So don't talk to me about fair," Murray exclaimed. Realizing he had gone a bit overboard, Murray proclaimed, "Hey, I want to help you out, so let me give you a tip: That balance scale may be used only once." "What? Only once?" she said, thinking out loud. "I wish I had another balance scale." ZAP! Another scale appeared. "Okay, kiddo, that was wish three," Murray snickered. "Hey, just one minute," Alley said, now regretting not having asked for one million dollars or something more standard. "Well at least this new scale works correctly, right?" "Sure, just like the other one. You may use it only once." "Why?" Alley inquired. "Because it is a 'wished' balance scale," he said, "so the rule is 'one scale, one balancing'; it's just like the rule against using one wish to wish for a hundred more wishes." "You are a very obnoxious genie." "Hey, I don't make up the rules, lady, I just follow them," he said. 0
So, Alley may use each of the two balance scales exactly once. Is it possible for Alley to select the slightly heavier stone containing the Rama Nujan from among the nine identical- looking stones? Explain why or why not.
Solution
Yes, Alley can determine the heavier stone (the one containing the Rama Nujan) from among the 9 identical-looking stones using the two balance scales exactly once. Here's how:
Key Idea:
Since there are only two weighings allowed and Alley has 9 stones to choose from, the challenge is to divide the stones in a way that allows her to identify the single heavier stone through a process of elimination.
Step-by-Step Solution:
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Divide the Stones: Alley can divide the 9 stones into three groups of 3 stones each. Label the groups as Group A, Group B, and Group C.
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First Weighing:
- Place Group A (3 stones) on the left side of the first balance scale, and place Group B (3 stones) on the right side of the scale.
- Now, there are two possible outcomes:
- If the scale balances (both sides weigh the same), then the heavier stone must be in Group C (the remaining 3 stones).
- If the scale tips (one side is heavier), then the heavier stone is in the heavier group (either Group A or Group B).
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Second Weighing:
- Now, after the first weighing, we know the group that contains the heavier stone (either Group A, Group B, or Group C).
- Take the 3 stones from the heavier group and place two of them on the second balance scale (one on each side).
- Now there are two possible outcomes:
- If the scale balances, then the third stone (which was not on the scale) is the heavier one, and thus the Rama Nujan is in that stone.
- If the scale tips, then the heavier side contains the stone with the Rama Nujan.
Why This Works:
- After the first weighing, we eliminate 6 stones (the ones in the group that didn't contain the heavier stone).
- After the second weighing, we find the heavier stone by comparing only 3 remaining stones, which is a manageable number to narrow down the possibilities.
Thus, using this method, Alley can identify the heavier stone with the Rama Nujan, following the rule of using the scales only once each.
Conclusion:
Yes, it's possible for Alley to find the heavier stone, which contains the Rama Nujan, by dividing the stones into three groups and using the two balance scales exactly once.
Would you like more details on how this strategy works, or perhaps how to approach similar problems?
Here are some related questions:
- How many stones would you need to solve this problem with only one weighing scale?
- How would you adapt this method if you had more than two balance scales?
- How could this method be used if there were 12 stones instead of 9?
- How does the principle of dividing into smaller groups relate to binary search methods?
- What would happen if you had more than one heavier stone? How would the problem change?
Tip: When you are working with limited resources (like a limited number of weighings), dividing the problem into smaller, manageable sections is often the key to solving it efficiently.
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Math Problem Analysis
Mathematical Concepts
Problem Solving
Logic
Weighing and Elimination
Formulas
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Theorems
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Suitable Grade Level
Grades 6-8