ThreeRings for the Elven-kings under the sky,
Seven for the Dwarf lords in their halls of stone,
Nine for Mortal Men doomed to die,
One for the Dark Lord on his dark throne
In the Land of Mordor where the Shadows lie.
One Ring to rule them all, One Ring to find them,
One Ring to bring them all and in the darkness bind them
In the Land of Mordor where the Shadows lie.
Everyone who has read the Lord of the Rings is familiar with the poem above. I am still making my way through the Silmarillion and will eventually get to reading the Unfinished Tales, which may hold the answer to my question: Is there any significance to the number of rings given to each race or is it just based on the number of kings/lords who ruled each race? Perhaps there is some information somewhere of why Tolkien chose these odd numbers or why the total number of rings is 20? Or perhaps there is no significance at all...
The Three Elven Rings are noted in Of the Rings of Power and the Third age as having been made last, and that Sauron had no part in their making (despite which they were still subject to the domination of the One):
Now these were the Three that had last been made, and they possessed the greatest powers. Narya, Nenya, and Vilya, they were named, the Rings of Fire, and of Water, and of Air, set with ruby and adamant and sapphire; and of all the Elven-rings Sauron most desired to possess them, for those who had them in their keeping could ward off the decays of time and postpone the weariness of the world. But Sauron could not discover them, for they were given into the hands of the Wise, who concealed them and never again used them openly while Sauron kept the Ruling Ring. Therefore the Three remained unsullied, for they were forged by Celebrimbor alone, and the hand of Sauron had never touched them; yet they also were subject to the One.
An important thing about the others is that's there's actually nothing in Tolkien to indicate that the powers of the nine are any different to the powers of the Seven; rather, what differentiates them is the species they were given to, with a Ring of Power having a different effect on a Man to the effect that it has on a Dwarf.
There Sauron took the Nine Rings and other lesser works of the Mrdain; but the Seven and the Three he could not find. Then Celebrimbor was put to torment, and Sauron learned from him where the Seven were bestowed.
So the Nine Rings that Sauron gave to Men were just the first nine that he had captured, and after having found where the remaining Rings (excluding the Three) were located, those were the ones that he gave to Dwarves.
There's further room for some interesting speculation which it's probably worth mentioning to ward off follow-up questions. We know that there were three houses of Elves and Seven houses of Dwarves, which match with the numbers of Rings each had. Were there also nine houses of Men (in total, including the three of the Edain that we know of for certain)? There's nothing in Tolkien to suggest this.
The Rings of Power were twenty magical rings forged in the Second Age, seventeen of which were intended by Sauron to seduce the rulers of Middle-earth to evil. Disguised as the benevolent entity Annatar, Sauron taught the Elf-smiths of Eregion, led by Celebrimbor, how to craft these rings. Nineteen were made: three rings for the Elves, seven rings for the Dwarves, and nine rings for Men. An additional ring, the One Ring, was forged by Sauron himself at Mount Doom. The nineteen lesser Rings were linked to the power of the One, and were dependent on it. Their wearers could be controlled by the wearer of the One, and if the One was destroyed, their own powers would fade with the power of the Rings under the One.
The Rings of Power were the masterwork of the Gwaith-i-Mrdain of Eregion headed by Celebrimbor, grandson of Fanor. The impetus for their creation came from Sauron, who could at that time still assume an appearance fair enough to deceive at least some of the Elves. The concept of these Rings of Power were intended to slow the "decay", preserving the world, so that it resemble the beauty of Valinor. [1] A total of nineteen Rings of Power were forged by the Elves, sixteen of which Sauron had a direct hand in creating. The greatest three Rings Celebrimbor crafted alone. Many other lesser rings were made, described in The Silmarillion and by Gandalf, though they were generally considered as having been mere essays in the craft: practice, as it were, for the smiths. Sauron, however, planned to use the Rings to dominate the remaining Elves of Middle-earth. In accordance with this, he secretly forged the One Ring in the fires of Mount Doom, seeking to bring all the rings and their wearers under his sway. However, when Sauron put the Ruling Ring on his finger, the Elves were immediately aware of him and took off their Rings. Furious at this turn of events, Sauron came against the Elves with open war and demanded that the Rings be given to him. Fortunately for the Elves, they were able to hide the greatest Three and pass one to the Dwarves of Khazad-dm, but Sauron recovered the other fifteen. These he then gave to mortals: the Dwarves and Men. Six he gave to Dwarves (for Seven in total), but Nine he gave to Men, knowing that they would most easily bend to his will. In time, the nine Men to whom the Rings were given became Ringwraiths, or Nazgl, wraiths of great power under the command of Sauron. The Dwarf lords to whom the Seven were given became wealthy beyond measure, but they did not fall to Sauron's power. They were essentially too "tough" in spirit to be dominated by the will of another, even Sauron. However, the Rings kindled in them an overwhelming greed for treasure, and so ruin was brought to each regardless.
According to Gandalf, one feature of the Rings of Power that appeared to be universal was that a mortal who wore any one of the Rings was granted a lifespan far beyond their natural one. It is likely that this is an unintended side-effect of the Rings' power, as Sauron originally intended for all the Rings to be worn by Elves, who were immortal to begin with. The individual who kept the Ring would not grow or obtain more life, however; they would merely continue until living became unendurable. Bilbo Baggins stated near the end of his time possessing the One that he was beginning to feel "thin and stretched", a sign that this process was beginning to affect him.
The Three which remained to the Elves were carefully hidden from Sauron. They were not created as weapons of war or as a means to dominate others; their purpose was to preserve the beautiful Elven domains where their bearers resided, and to aid in healing and resisting evil. However, they were still linked to the One Ring, and whoever wore the One could, with effort, see the thoughts of those who wore the Three. As such, these Rings were not used by the Elves so long as Sauron retained the One.
In The Silmarillion, it is indicated that the Seven Rings of the Dwarves were not fundamentally different from the Nine that were given to Men, for originally Sauron had intended both the Seven and the Nine to reside in Elven hands. It is implied in that work that Sauron simply gave Men nine of sixteen lesser Rings because they were more easily controlled. However, the Nine and the Seven are referred to in other works as each being its own distinctive set. If they were different in their effects on their wielders or in the powers they conferred, it is not stated. It is also mentioned in Unfinished Tales that the Seven were created before the Nine. Sauron helped to create the Seven and the Nine, so it is likely that their powers and effects on their wielders were similar, even if each set was distinct. It is important to note that the fact that the Seven did not subdue the Dwarves is related to the resilient nature of the Dwarves themselves, not the Rings. It is likely that the effects of one of the Nine on a Dwarf would be the same as those of one of the Seven, and were a Man to wield one of the Seven, he would likely still become a wraith. The Three were fundamentally different because Celebrimbor, working alone, created them for specific purposes not in tune with Sauron's line of thought, and Sauron presumably did not know of their creation until after the fact.
As told in the works of The Silmarillion, the Elves made many other Magic Rings, but these were mere practice for the craftsmen who created them. Their ultimate fates are untold, as are their connection to the One Ring, if any.
Celebrimbor forged the Three Rings by himself, and they were never touched by Sauron. They were called Narya (the Ring of Fire), Nenya (the Ring of Adamant), and Vilya (the Ring of Air). Narya was first worn first by Gil-galad, who gave it Crdan, and was lastly worn by Gandalf. Gil-galad also initially owned Vilya, before Elrond inherited it after the former's death. Nenya was worn only by Galadriel.
They remained hidden, and the whereabouts of the three were not openly revealed until the end of the Third Age, after the One Ring was destroyed, and the Dark Lord Sauron was overthrown. Instead of making the wearer invisible, these rings were themselves invisible. Galadriel revealed her possession of one of the rings to Frodo Baggins while he was in Lothlrien, after his possession of the One Ring enabled him to see her ring. Later, while traveling down the River Anduin, Frodo spoke of it to Aragorn, who admonished him not to speak of it outside of Lrien.
Each of the Three Rings had special properties, but their powers were limited. During the Council of Elrond, Elrond stated that neither Rivendell, nor Lrien, nor the Grey Havens (the locations of the Three Elven Rings) had the power to withstand the might of Mordor.
According to Unfinished Tales, at the start of the War of the Elves and Sauron, Celebrimbor gave Narya to Crdan, Lord of the Havens of Mithlond, who kept it after Gil-galad's death. In the Third Age Crdan, recognizing Gandalf's true nature as one of the Maiar from Valinor, gave him the ring to aid him in his labors. It was described as having the power to inspire others to resist tyranny, domination, and despair (in other words, evoking hope from others around the wielder), as well as giving resistance to the weariness of time:
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