The cool breeze raising from the Zeravshan river awoke the mathematician on the morning of the third day. Al-Khwārizmī skipped his breakfast and went directly to his worktable in the royal garden, where, according to his instructions, new loads of paper were already in place. As the hours of the day passed by, the initially light mood of Al-Khwārizmī progressively turned somber, and his writing frantic; around him, scattered on the floor, lay dozens of sheets of paper full with garbled diagrams and tables.
This routine went on during the following days, and the King, a little impatient about the outcome of al-Khwārizmī's quest, began to send messengers to secretly observe the mathematician from the distance. This would not talk to anyone and abruptly dismissed the servants who approached him with food and wine. He had some lights set up at his place on the veranda so that he could continue working during the night.
One chilly morning Al-Khwārizmī found that he had fallen asleep at his workplace. He then decided that the quest was over and asked for an audience with the King. After explaining to the sovereign how he devised the principles of his theory for a Philosopher Rule, the mathematician then concluded:
"I have spent several days and nights searching for suitable lists of authoritative groups, beginning with some initial disposition and adding and supressing groups according to the logical principles that I have presented. And anytime I obtained a distribution conforming to the principles, I realized to my dismay that it possesed the following defect: there was some particular philosopher such that a group was authoritative in as much as that philosopher belonged to it. If I built some other suitable different distribution, it was only that particular philosopher that changed, but the troublesome peculiarity remained. What is the usefulness of a Rule if it merely takes into account the opinion of a single philosopher in the Council?"
"My lord, in despair and humiliation I must resign from the task that you gratiously assigned me. As far as my investigations have led me to see, there is no way to design a logical Philosopher Rule other than following the decisions of a single designated individual in your Council."
"Great al-Khwārizmī" replied the King, "it is a true sign of the wisest to acknowledge what one does not know. And you have my admiration for this sincerity and for the fine mathematical work you have presented to me."
"Given that your investigations show that having a Council of philosophers is no better than relying in one of them, I can not think of any other man than you who is more suitable for that position. Would you accept my humble invitation to be the Royal Counselor?"
In the end al-Khwārizmī declined the proposition and promptly left Samarkand to go on his trip to remote countries. History has it that the King dismantled the Council and devoted himself to the study of Science and Philosophy so as to become a better person, a wiser ruler, and his best own counselor.