Once upon a time, there was a kingdom so vast and so wide that the kings of the surrounding empire sought for control. Now this land was not only vast, but it was the home of an enchanted queen, who had been blessed and cursed by a witch. Blessed, for her kingdom and her land would forever flourish in the hands of the ruler. Cursed, for her castle would forever be plagued with monstrous beasts as her servants. Cursed, for the queen herself would forever be confined within her tower, for the enchantments that surrounded her home were far and many.
Yet the kings of the surrounding magical land sought to claim the hand of the queen and the land of enchantment. For whosoever retained ownership of such a land -- and whosoever married such a queen -- would also be blessed beyond imagining.
Not everything, however, comes without a price. Whenever a neighboring ruler sought the hand of the queen, his failure would prove fatal, and the land in which he once ruled would dissolve and become part of the queen's land. It must have been in this manner that the queen's land doubled in size, for king after questing king failed in such an attempt, and his people fell under the enchanted lands. The cycle would continue for years on end, with the queen never aging and always cursed.
Now far off into a distant land, a prince heard the stories of the queen's curse. While he also heard of the blessings of the land that surrounded the cursed queen, he lamented the woman's problems, and was heartbroken at her seemingly unending curse. For this prince, kindly and goodly that he is, did not care for riches and lands. His own father taught him that the land was only as good as the ruler, and when ruled by a most benevolent and wizened hand, it would flourish without enchantment. It was the plight of the queen that stirred the prince's heart. And so it was with this goal in mind that the prince planned to set out.
The prince's father, an aging king with no other offspring, pleaded for his son not to carry out such a quest, for the father also knew what would happen to those who failed. But the prince was adamant, and the more he tried to forget, the more the queen's plight burned into his mind, and days passed where he could neither eat nor sleep until his father allowed him this permission.
Before the prince went off to carry out his journey, the old king bestowed upon him five of his bravest and most gifted knights. The king was also not without any magic of his own, and he entreated a sorcerer to give the prince a magical item, with which he could divide up between his knights. Employed by the king, the sorcerer took off his long cloak and handed it to the prince.
"This cloak has many different purposes and enchantments," said the sorcerer. "Divided, the power can benefit you and your knights on this journey."
Then the king handed his son a ring from his right finger.
"Son, I gift to you a ring that has kept my mind clear and calm all these years," said the king. "May it aid you as it did me in my lifetime."
The prince thanked the sorcerer and bade his father farewell. And away he went toward the distant and vast land of the enchanted queen.
It was in this journey that the prince's wisdom became tested by his five noble knights. After a day's journey, the prince and his knights sat amongst themselves around a campfire in a forest clearing, and they immersed themselves in deep conversation.
"The sorcerer has advised us to divide the cloak, my liege," said the first knight. His other companions nodded assent. "Will you do so fairly, prince?"
Suddenly, the knights clamored up, demanding that each deserved a greater share than his compatriot. The clamor became a wild noise, and it unnerved the prince, for the noise might attract a greater evil within the forests that they lay camped. The prince fiddled with his ring, and suddenly the solution became as clear in his eyes as that of freshly-blown glass.
He stood and brought the argument to a hushed noise, for while the knights watched each other with wary eyes, they respected their master's son.
"Each of you but one shall divide this cloak into fair shares amongst yourselves," said the prince. "And once you have divided it amongst yourselves, you shall divide your own shares into five fair portions."
"What shall the fifth knight do, our prince?" asked the knights in unison.
The prince nodded. "The first knight shall choose a share from each of your divided shares, and that, I decree, will be your equal share of the cloak."
The knights wondered at the wisdom of this, but they chose to concede. The cloak glimmered with patches of stars, swirls, midnight blues, and deep grays, some of which were favored by one or two of the knights, and others which were not favored at all. The first knight liked the patterned swirls and stars just fine, but would have preferred the grays over the blues. The second knight didn't care what portion he received, only that he received a fair share. The third and fourth liked the colors just fine, but the third liked the stars more and the fourth liked blue and either pattern. The fifth knight didn't like swirls.
And so the division came along between the last four knights as follows:
1 The four knights divided the cloak into four shares as seen in Figure 1. The upper-left share went to the third knight, the upper-right went to the fourth knight, the bottom-left and the bottom-right went to the second and fifth knight, respectively. With this split effectively in place, the knights were then asked to split their own shares into five fair parts. Because of each knight's preference, this split was done differently for each knight.
Once the pieces were divided on their own shares, the knights placed value upon each subshare. The subshares, thus, looked similar to the following table:
Table 1. BIDS SUBDIVISION
Now, the first knight also placed a value on each of the pieces, and because he was the chooser, he began to take pieces from each knight. From the first divider, he chose Piece 2, from the second he chose Piece 2, from the third Piece 5, and from the fourth Piece 1. The cloak was then cut up in fragments and divided, with the second knight getting 80% of his subshares, the third knight getting 70% of his subshares, the fourth getting 82% of his subshares, and the final knight getting 90% of his subshares. In accordance to the bid shares given by the other knights, the first knight landed with 78% of subshares.
Happy with this arrangement, the knights sewed the pieces of their enchanted cloaks into their own capes. Within minutes, they felt their strength and skill grow leaps and bounds. With a happy jump, the first knight proclaimed his gift in speed, and in-so-doing, he spent all of the first night strengthening and motivating the horses to travel at the pace of the wind. Within the next day, their journey was considerably sped up, and they saw the land of prosperity within their sight.
But first, they had to travel down a steep and treacherous mountain. So it was with pleasure that the second knight proclaimed that he was gifted with super strength, and in-so-doing, tied the ropes of all the horses to his own, and remained at the back, keeping the prince and the other knights from falling too quickly at the steep climb down.
As they reached the entrance to the castle, they were once again impeded by a large moat, almost as wide as a river. It was the third knight's turn to proclaim that his gift allowed him to swim well and breathe for long periods of time underwater, and so it was with this that he plunged into the moat, emerged at the other side, and pulled down the menacing drawbridge.
The monsters approached them then, their breaths filled with smoke and fire. The fourth knight, gifted with a resistance to the elements, stood at the helm, charging against the monsters and parting the way for the prince.
There were two obstacles left to reach the tower in which the enchanted queen was confined, and in order to get to the tower, one had to climb from the outside, for there were no doors at the base. It was here that the prince…