**Problem Solving Story
**

**That’s a meanie genie!**[youtube https://www.youtube.com/watch?v=2tfXdjw1aPU]

On an archaeological 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 oppor- tunity. 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. How will 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.

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.

*Clue*

- Whole Numbers (Division)
- Measurement (Mass)

- Thinking skills
- Comparing
- Analyzing (from whole to part)
- Deduction

- Heuristics
- Simplifying a problem

*Instead of comparing stones individually, perhaps she should compare one collection of stones with another collection of **stones. Now suppose Alley compares one group with another using the first scale. What can she conclude? What should she do next?*

*Take partial steps whenever possible. Notice that, instead of trying to identify the jewel immediately, first reduces the pool of choices to make the problem easier. “Divide and conquer” is an important and useful technique in both mathematics and life.*

[youtube https://www.youtube.com/watch?v=_hqBxle90IY]

**Solution**

Alley arranges the stones into three groups of three and places one group on one side of the first balance scale and another group on the other side. What can she conclude? If both sides weigh the same, then she knows that the (heavier) jewel must be in the third group of three. If, however, one side is heavier than the other, then she knows that the jewel is one of the three that weighed more. In either case, after only one weighing, Alley is able to identify a group of only three stones among which is the Rama Nujan.

She then takes two of these three stones and places one on each side of the second scale. If one weighs more than the other, then she knows that this stone is the one containing the jewel. If they both weigh the same, then she knows that the third stone must contain the jewel.

Thus, by weighing the stones only twice, Alley is able to find the jewel.