The Nine Pearls

Abdul Hadi Badi-al-Zamar, the Sufi, arrived at the palace and was ushered in to see the princess.
"Come sit with me a while," she suggested, "and we will talk of the mysteries of life." They had known each other since she was a little girl, and Abdul Hadi admired her mind and delighted, to a degree that was proper, in her beauty. She displayed great promise as a mathematician.
"I have a problem for you," she said.
She clapped her hands together and her slave girl came in. "Bring the casket," she ordered, which the shapely young black woman did.
The princess sat up on her divan, opened the casket and poured its contents into a bowl, which pinged to the sound of falling pearls.
The slave girl, who must have known what was required, brought a pair of scales.

Painted by José Tapiró y Baró

"The problem is this," she said to the old man, "one of these pearls is false. They all look identical, but one is fractionally lighter than the rest, and is not so valuable. Using just the scales and the pearls, what is the least number of weighings necessary to find the false pearl every time?

The Sufi rubbed his beard, and let the pearls trickle through his fingers, one by one, back into the bowl.
"With luck, one weighing would do it, but to be sure every time, then ..."

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So, using just the scales and the pearls, no weights, how many times must the pearls be weighed to get the false pearl every time?

Abdul Hadi Badi-al-Zamar took the nine pearls and laid one aside, carefully.
"If this is the false pearl," he said, "we will know straight away. The other pearls, my Lady, are exactly alike you say?"
The princess nodded.
"If I divide them four by four, and they weigh the same, four in each pan, then this pearl," he pointed to the one in the bowl, "will be false."
She clapped her hands. "Of course," she said.
But the pan on the right was lighter than the pan on the left.
"Ah," said the Sufi, "then we have found that our false pearl is one of these four in the left hand pan."
The princess nodded and popped a grape into her mouth.
The Sufi divided the four pearls from that pan two by two and weighed them agaisnt each other. This time the right hand pan was lighter.
"Now we know it is in this other pan. And we will have our false pearl with one more weighing."
Taking the two pearls from the right pan and placing one each on the sides of the scale, the false pearl was immediately discovered.
The princess pouted. "I thought it would take you longer to do this calcualtion, my dear friend," she said. "But you are right. three weighings is all you need."
Abdul Hadi Badi-al-Zamar, smiled.
"Did I tell you about the time I came across three brothers arguing about their inheritance?"
The princess shook her head.
So he told that story which can be found here.