The Magic Number
Manu, a king in a long-forgotten land called Lemuria, that flourished prior to recorded history, had a nubile daughter named Lailai he decided to marry off to someone brave and intelligent to ensure his lineage would continue with prosperity.
So he had a proclamation posted on all trees and buildings in the kingdom that whosoever could work out the magic number would marry Princess Lailai. Those who failed would have their head chopped off.
According to the court astrologer Bolbol, the magic number was the test that would endow the person who could work it out with the power of dominion of Lemuria. The number was the total area of the kingdom, expressed in mets, the area unit used in Lemuria. Three digits were missing in the number.
The future prince would have to work out what these three digits were to qualify for marrying Princess Lailai and inheriting the kingdom after King Manu decided to leave his body for other regions.
The threat of death deterred all but three applicants, the first two of which were rapidly despatched by Zok, the executioner. The last candidate was Manas, the son of a blacksmith of a small town in Lemuria.
He had always had a thing with numbers and would dream about them all day long – even while working in his father’s smithy. Manas would write down his discoveries in a notebook during his free time.
Manas saw the proclamation on a tree he often conversed with, and was his special friend. He remembered having seen Princess Lailai in a procession, when a glow suddenly appeared around her head.
Next day, Manas, with his notebook, presented himself as a candidate before King Manu in the presence of Princess Lailai, who looked at him with large eyes.
Manas was led to a room in a turret by Bolbol the court astrologer – accompanied by a leering Zok with his axe – given writing materials and told to submit the answer to Bolbol the next day after breakfast.
The magic number 8*45*64*9 was written on a board on the wall of the turret room. Bobol said it was divisible by 9 and 11, and was a square. Then he left, locking the door behind him.
Initially, Manas had some difficulty in working out the number, but then he remembered something he had discovered about the endings of squares and written down in his notebook, which made the whole thing much simpler.
After breakfast the next day, Manas submitted his answer to Bolbol, the court astrologer, who confirmed that it was correct.
Soon thereafter, Manas and Lailai rode off on two royal stallions of the time to see the countryside. Standing on a balcony, King Manu watched them briskly ride off with a smile.
What do you think was the magic number Manas worked out so he could marry Princess Lailai?
Tags: congruences, divisibility, modular math